Quadratic Function Minimum Or Maximum Value Calculator

Given two points on the graph of a linear function, we may find the slope of the line which is the function’s graph, and then use the point-slope form to write the equation of the line. Graphically, you’re looking for a global maximum. Since the coefficient of #x^2# is negative the parabola would be open down. If I have a maximum or a minimum, and we'll have pictures, somewhere in the middle of my function I'll recognize by derivative equals 0. We know that linear equations graph a straight line, so I wonder what a quadratic function is going to look like?. How To Find The Maximum And Minimum Values Of A Trigonometric. linear equation 2. Example 1 : Find the coordinates of the vertex of. CASIO GRAPHING CALCULATORS TI GRAPHING CALCULATORS Finding the Max and Min of a Function continued CALCULATORS: Casio: fx-9750G Plus & cfx-9850G Series. This video looks a finding the minimum or maximum of a quadratic function. (a) f(x) = 3x2 ¡6x+2 Here, we know that a = 3; b = ¡6 and c = 2. Determine the domain and range of the function f of x is equal to 3x squared plus 6x minus 2. A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The Domain for a quadratic function is all real numbers. EDIT I guess you could say that the function (defined on (0,10]) has a local minimum value at x = 10 because it is an endpoint though. • The factored form of the function is the best for identifing a quadratic function’s roots. Math 2 – Quadratic Functions QUIZ 5. 3 Ways To Find The Maximum Or Minimum Value Of A Quadratic Function. The following quadratic equation in vertex form. Then differentiate using the well-known rules of differentiation. For the factored form, f (x) (ax b), the midpoint. notebook February 22, 2016 Determining Maximum and Minimum Values of a Quadratic Function Learning Goals ­ recall how to convert a quadratic function to vertex form by completing the square ­ determine four ways to find the max /min of a quadratic function. Finding solution oredered pairs caculator, radical equation calculator, 1, 4th grade find factors, odd function with points, rational expresssion applications. Study Helper The Student Room. If a quadratic function’s vertex is on the x-axis, then it has exactly one solution. We substitute this x-value into our quadratic function (the y expression). Determine whether the given quadratic function has a minimum or maximum value. Only three sides must be fenced, since a rock wall will form the fourth side. A quadratic function has two real solutions. Range: write range to reflect the maximum or minimum value of 'y'. minimum of y = 5 (1 each) d) Calculate the y-intercept. Then find the coordinates of the minimum or maximum point. Q: A farmer wants to put a fence around a vegetable garden. To Graph a Quadratic, enter the following key strokes. Find the minimum value of Rosenbrock's function when there is a linear inequality constraint. 3 ways to find the maximum or minimum value of a quadratic function 3 ways to find the maximum or minimum value of a quadratic function 3 ways to find the maximum or. Axis of Symmetry; Maximum and Minimum Value of Quadratic Functions. If , then has a global minimum, and no global maximum. Then, plug this value back into the function. 5 Deriving Quadratic Functions 239 1. Markscheme (i) (M1) Note: Award (M1) for correct substitution in formula. the maximum and minimum value of f occurs at x = h. Graphing Quadratic Equations. The y-coordinate of the vertex is the minimum y-value of a parabola that opens upward. If the function is f(x)= a. Answer key Score : Printable Math Worksheets @ www. Example 2: Determine the minimum value of each quadratic function and the value of x. You need to notice that the equation of provit function is a quadratic equation and since the leading coefficient is negative, the maximum value of the profit function `y = (4ac - b^2)/(4a)` is. y = 2(-2)^2 + 8(-2) + 5 = -3. We wish to MINIMIZE the DISTANCE between points (x, y) and (4, 0) ,. The graph of a quadratic function is a specific kind of curve called a parabola, a sort of U-shaped figure. The y-coordinate of the vertex is the minimum valueof f. They are called local or relative maximums or minimums because, compared to other nearby points, they have the largest or smallest y-value. Maximum and minimum values of a quadratic polynomial We will learn how to find the maximum and minimum values of the quadratic expression a x 2 + b x + c , a ≠ 0. Then find the minimum or maximum value and determine where it occurs. This function is linear, not quadratic. The graph of a quadratic function is a curve called a _____. function or a decreasing function. Without a calculator, determine the vertex for each graph and state if it gives a maximum or a minimum value for y. decreasing on the other side. MATH TERMS. Find its maximum or minimum and determine the roots of the function. Therefore, even without graphing the quadratic function, we can find the extremes of a quadratic by plugging the coefficients a,b,c into the formula. Definition 1. Investigate quadratic functions in vertex form in this algebra lesson. of the axis of symmetry and. Deo b) c) a) D70. from this quantity he sells 80 scoops at $0. Algebra Examples. • The standard form of the function makes it easy to see the function's y-intercept or "starting value. The y-coordinate of the vertex is the minimum y-value of a parabola that opens upward. The minimum value of a quadratic function is the low point at which the function graph has its vertex. If the house serves as one side of the kennel, determine the greatest. If the coefficient of x 2 is positive, you should find the minimum value. To link to this page, copy the following code to your site:. Find the maximum and minimum value of the function possible when x is varied for all real values possible. The value of the minimum is. If the quadratic function is set equal to zero, then the result is a quadratic equation; the solutions to the univariate equation are called the roots of the univariate function. Max Min Word Problems So we use our techniques for nding the maximum value of a function on a closed interval when the function is continuous there. If we take the second derivative and if that value is positive, then we are dealing with a minimum value. In this activity, you will examine how the shape of the parabola changes as the values of a, b, and c are modified. 678(72)2 + 527. a x 2 + b x + c , a = 0. The output will be your maximum/minimum. the minimum value of y occurs at the origin. The interval can be specified. Students will graph quadratic functions using input/output tables and technology. But first, we need to examine one additional quadratic function by hand. 2 Quadratic Approximations To get a better approximation of our function we try to approximate it using a quadratic polynomial. 3: Polynomials and Rational Functions Sec. The maximum or minimum value of a quadratic function occurs at the vertex and is represented by the y-coordinate Of the vertex. Emphasize to learners the importance of examining the equation of a function and anticipating the shape of the graph. In this case, the relative maximum is located at 1 x 2 and the relative minimum is located at x 3. Sketch the graph of a quadratic and absolute value function. (a) If one of the sides of the rectangle measures 'x' cm, find a formula for calculating the area of the rectangle as a function of 'x'. (Use your graphing calculator to find the coordinates of the vertex. This algebra video tutorial explains how to solve word problems that asks you to calculate the maximum value of a function or the minimum value of a quadratic equation. Let's Practice:. 1 Properties of a Quadratic Function—Solutions DO NOT COPY. Therefore if you want to know the maximum revenue (and the associated price to get that maximum revenue), you are asking to find the vertex of the parabola. (d) If f xx x 2312 , then find, without using your calculator, the value of f 2. The graph of a quadratic function contains the point (0, 0). Min/Max of a Quadratic Function. A graphing calculator is recommended. have a relative maximum at the critical value. a) State the coordinates of the vertex. CHAPTER 11: Quadratic Equations and Functions calculator corner Maximum and Minimum ValuesWe can use a graphing calculator to find the maximum or minimum value of a quadratic function. Once you have the roots, you can just simply plug in the x value to the original function where y is the maximum or minimum value. This may explain the difficulty in applying the usual test. How to calculate a maximum of a function? The maximums of a function are detected when the derivative becomes null and changes its sign (passing through 0 from the positive side to the. The Interface. Because the derivative provides information about the gradient or slope of the graph of a function we can use it to locate points on a graph where the gradient is zero. One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function. Students will identify and describe the effect of an equation’s parameters on its graph. Here’s one less common proof that [math]x=-\frac {b}{2a}[/math] is the value that maximizes or minimizes a quadratic function: Let [math]f(x) = ax^2+bx+c[/math]. To solve real-life problems, such as modeling the height of a sunflower in Example 5. The maximum/minimum are given by x=-b/(2a). Apart from the stuff given above, if you want to know more about "Maximum and minimum values of quadratic functions worksheets ", please click here. If a < 0 in standard form, factored form, or vertex form, then the. A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0. To find these important values given a quadratic function, we use the vertex. In this unit, we will study quadratic functions and the relationships for which they provide suitable models. 2: Determining Maximum and Minimum Values of Quadratic Functions The maximum or minimum value of a quadratic function is the _____ of the vertex. A quadratic function’s minimum or maximum value is given by the y-value of the vertex. x-intercept A. Drawing a triangle from the viewers eye to the pin, up the pin, and back to the viewer's eye; and drawing a 2nd triangle from the viewers eye to the edge of the viewfinder up to the apparent height of the pin (the image) in the viewfinder and back to the. m will contain the primary function quadratic and the sub-function disc, which. You also need to write a procedure to call this function. Now the "K" value, +1 is your maximum value. To find the minimum it is simply lowest y coordinate. It is often useful to find the maximum and/or minimum values of functions that model real-life applications. Once you find the a. Give the formula to calculate R^2 in terms of Sxx, etc. logistic growth functions. First we need to choose an interval:. Determine a quadratic function’s minimum or maximum value There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue. Don’t forget to draw the axis of. Maximum and minimum values of a quadratic function. In this problem we have-----> this a vertical parabola open upward. Listed below are word lessons that focus on giving students instruction on how to solve most types of word problems commonly found in algebra, geometry, and trigonometry. They will identify the vertex, axis of symmetry, maximum and minimum, roots, zeros, solutions, x-intercepts, y-. If the second derivative is negative (less than zero), then the calculated "x" or whatever variable will give be the max value that you are looking for. (b) Find the minimum value of 𝑓. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. And the absolute value of A is going to be equal to 1 over 4C. The graph of a quadratic function is a curve called a parabola. For each of the following. Use your calculator to check your answers. Therefore if you want to know the maximum revenue (and the associated price to get that maximum revenue), you are asking to find the vertex of the parabola. Use derivative rules to take the derivative of the function 2. If is negative, the maximum value of the function is. 70, there is a maximum value at the vertex. If , then , and therefore. Using this quadratic calculator or quadratic equation solver, we can find the following characteristics of a quadratic equation. First, we replace w with x and A with y and graph the function in a window that displays the vertex of the graph. The maximum or minimum of a quadratic function occurs at. Algebra 1, Unit 5 – Quadratic Functions – L1 The Arlington Algebra Project, LaGrangeville, NY 12 540 When the parabola opens up, the parabola has a minimum y-value and when the parabola opens down, the parabola has a maximum y-value. Recognize the Graph of a Quadratic Function. Students use the CBR (Calculator Based Lab) and a graphing calculator to collect data from a bouncing ball, graph the data, and model the curve with an equation. The lessons in this unit focus on working with quadratic relationships. 5 use a graphing calculator and plug indoors the function the vertex is on the line of symmetry that's additionally the max/min so press hint and enter indoors the line of symmetry,. Quadratic Equations. The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. Study Helper The Student Room. Then differentiate using the well-known rules of differentiation. 3 Ways To Find The Maximum Or Minimum Value Of A Quadratic Function. Find more Mathematics widgets in Wolfram|Alpha. Extreme Values for Quadratic Functions For more details on this, see the handout on quadratic functions. 1 Understand that a function from one set (the input, called the. a change in a function rule and its graph. The basic parabola equation is given as a function:. Minimum or Maximum Values of a Quadratic Equation The y-coordinate of the vertex of the graph of a quadratic equation is the; minimum value of the quadratic equation if the parabola opens upward. m on your MATLAB® path. As you make. So that's actually one way we can figure out exactly how steep or. A rule of thumb, if a is positive, the function has a minimum. If the coefficient of 'a' in the quadratic form of the equation is positive the vertex will be. As discussed in the previous article, the vertex is either maximum or minimum of the parabola. In this problem we have-----> this a vertical parabola open upward. Write expressions in equivalent forms by completing the square to convey the vertex form, to find the maximum or minimum value of a quadratic function, and to explain the meaning of the vertex. To know if its a maximum or minimum value, simply do your number. Make a conjecture about the relationship between the vertex of a quadratic function and the values of a, h and k in the vertex form of a quadratic equation. To find it, calculate the coordinates of the vertex. Graphs of Quadratic Functions Parabolas Minimum Vertex Axis of symmetry Maximum 5. Lesson 6-1 Graphing Quadratic Functions 289 b. The minimum value of the function is the y-coordinate of the vertex. Through simple Excel 2010 functions you can easily find out the maximum and minimum value in the spreadsheet. If the leading coefficient a is positive, then the parabola opens upward and there will be a minimum y-value. Minimizing Quadratic Functions 1. Find the maximum and minimum value of the function possible when x is varied for all real values possible. This point is called the vertex of the graph of f. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape. x f(x) = x2 y (x,y) -2 -1 0 1 2 Verify your graph is correct by graphing the function on your calculator. ) The function y= x 2 is shown in the graph. Find the maximum and minimum value of quadratic function : The graph of the quadratic equation will always be a parabola which is open upward or downward. at x= -b/2a (absisca of vertex), so the minimum val. If students are not familiar with the correct term, they should still be able to describe its form as being roughly U-shaped. how do I find the relative maximum and minimum values of x^4-x^3-7x^2+x+6. Coefficients are numerical values. The quadratic has a minimum value. The gradient of the curve at any point is:. maximum value of the quadratic. Find the y-intercept, the equation of the axis of symmetry and the x-coordinate of the vertex. txt) or view presentation slides online. There can be two cases: Case 1: If value of a is positive. This algebra video tutorial explains how to solve word problems that asks you to calculate the maximum value of a function or the minimum value of a quadratic equation. If we wanted to figure out where does this function intersect the x-axis, what form would we put this. Transformations of quadratic functions The quadratic parent function graphs as a parabola: In the examples 1-6, begin with this parent function y = f(x) = x2 and graph the indicated transformation: Example 1: y = f(x) + 3 Each point of the new function is the corresponding point of the original parent function translated (shifted) up 3 units. Discuss the different functions and the effects of the parameters in general terms. clearly visible, and the maximum or minimum value of the function is k. The quadratic function with a < 0 has a maximum point at (h, k) and the function is increasing on the interval (-infinity, h) and decreasing over the interval (h, + infinity). b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. minimum value and then 7)f(x) =-x2-2x-4 Find the Vertex Is the vertex a minimum or maximum?. On the other hand, if the coefficient a < 0, then the parabola opens downward and the vertex is the highest point on the parabola. We can help you solve an equation of the form "ax 2 + bx + c = 0" Just enter the values of a, b and c below:. What selling price will produce the maximum revenue and what will the maximum revenue be? A:. b) Graphical representation of quadratic functions. A(7)(A) graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry. If a > 0, k is the minimum. The curve y=a. x+c is a parabola with its axis of symmetry parallel to the Y-axis and it opens upwards if a>0 and downwards if a<0. Graphing Quadratic Equations. So one of the applications of a quadratic equation is to find a maximum or minimum of a relationship and one of the most common relationships we're looking at is something being thrown up and then coming back down and looking for the maximum height and when that maximum height occurs. The form of a quadratic function can help you determine if the vertex of a quadratic equation will be a minimum or a maximum of the function. Step 1 - Take constant to another side, Step 2 - Take 8 common from LHS, Step 3 - Completing the square by adding and subtracting square of 1 inside the bracket, Step 4 - Write as perfect square, Taking root both side,. My question is related to how to find a local maximum and local minimum. 3 Maximum or Minimum of a Quadratic Function • MHR 5 Maximum or Minimum of a Quadratic Function Some bridge arches are defined by quadratic functions. txt) or view presentation slides online. The highest point over the entire domain of a function or relation. Make a conjecture about the relationship between the vertex of a quadratic function and the values of a, h and k in the vertex form of a quadratic equation. 8 R E A L L I F E R L Graphs of Logistic Growth Functions Use a. If a > 0, k is the minimum. plot vertex (h,k), then count the a value. Students will sketch the graphs of quadratic functions using parameters of an equation. Steps to graph Intercept form plot x-intercepts (p,0) & (q,0), locate axis of symmetry, Calculate minimum or maximum point on axis of symmetry. A function which has the general form y = f(x) = ax2 + bx + c is called the quadratic function, where a, b and c are real numbers, and a ≠ 0. Find the maximum or minimum value of the quadratic function by completing. Quadratic Functions and Determine the quadratic function whose vertex is ( -2, -5) and Determine whether the quadratic function 45 has a maximum or minimum value. The graph of a quadratic function contains the point (0, 0). First, I'm going to created a named range for the prices in this list. graph{2x^3-3x^2-36x-3 [-5, 7, -120, 150]} Indeed, there is a local maximum at x=-2 and a local minimum at x=3. Once you have identified the coefficients, you can use a formula to find the x-coordinate and the y-coordinate for the minimum or maximum value of the quadratic equation. (The x value will be the time. To form a quadratic equation when you are given the vertex, focus or directrix, click here: Calculating a Quadratic Equation from the Vertex, Focus or Directrix. a change in a function rule and its graph. Complete the following: The. This is a parabola opening downward. Line Equations Functions Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Minimum Calculator. • Solve applied problems involving maximum and minimum function values. The vertex can represent a minimum point or maximum point depending on whether the parabola opens up or down. Calculation of Maximum and Minimum Value 2a. The minimum value of a function is found when its derivative is null and changes of sign, from negative to positive. A quadratic equation is a polynomial equation in one unknown that contains the second degree, but no higher degree, of the variable. 8: Optimization of Functions of Several Variables - Mathematics LibreTexts. a positive coefficient before x 2 we have a minimum value, and if we have a negative coefficient we have a maximum value instead. Prerequisites. We can see where they are, but how do we define them? Local Maximum. They also rewrite expressions in vertex form to solve problems about the maximum or minimum value of a function. Quadratic Functions and Equations. How do find the minimum/maximum point on a curve? and can you give an example please By the way keep this to Core 1 level/AS level Exam board is edexcel. And the absolute value of A is going to be equal to 1 over 4C. If a > 0, k is the minimum. If the x^2 coefficient is positive, the function has a minimum. How To Find The Maximum And Minimum Values Of A Trigonometric. Links to examples and video for writing a quadratic function in standard form. Students find the vertex, axis of symmetry, solutions, roots, and graph quadratic functions. Don’t forget to draw the axis of. C program to find maximum and minimum using functions February 19, 2016 Pankaj C programming C , Function , Program Write a C program to input two or more numbers from user and find maximum and minimum of the given numbers using functions. Identify the function's domain and its range. → Read More: Max/Min Problem - Maximizing Revenue, Selling Calculators. Minimum Value Background • Minimum Value (MV) is the percentage of the total allowed costs of benefits provided under a group health plan or health insurance coverage, so long as that percentage is no less than 60%. Sketch the graph of a quadratic and absolute value function. 1 Find Maximum and Minimum Values - Practice - Page 244 4 including work step by step written by community members like you. This fun ction is often called the parent quadratic function. the maximum and minimum value of f occurs at x = h. • The graph of any quadratic function is a vertical and/or horizontal shift of a vertical stretch or shrink of the basic quadratic function f (x) = x2. Creating Transformational Form of a Quadratic (Completing the Square) Lesson 6. asked by student on June 24, 2013; Calculus. The highest point over the entire domain of a function or relation. That is to say: All you have to do is the arithmetic with your given coefficients. Use your calculator to check your answers. If a is negative, the function has a maximum. This tutorial takes a look at the maximum of a quadratic function. Creating Quadratic Equations. If I have a maximum or a minimum, and we'll have pictures, somewhere in the middle of my function I'll recognize by derivative equals 0. An-other thing we could try is to find a polynomial that has the same value as the function at some point a, the 16. They are called local or relative maximums or minimums because, compared to other nearby points, they have the largest or smallest y-value. Links to examples and video for writing a quadratic function in standard form. I cannot figure this problem out. Using the graphing calculator, graph each function. Hence the range of f is therefore given by the interval: (-∞ , 4 ] (see graph below to better understand) Fig2. You can put negative numbers if you need to use a negative coefficient. Get an answer for 'How do I find the minimum and maximum values of a function?' and find homework help for other Math questions at eNotes. Write expressions in equivalent forms by completing the square to convey the vertex form, to find the maximum or minimum value of a quadratic function, and to explain the meaning of the vertex. The quadratic calibration spreadsheet (Download in Excel or OpenOffice Calc. graph the function as well as its derivative using Wolfram|Alpha. We can see the maximum and minimum values in Figure 9. Here's one less common proof that [math]x=-\frac {b}{2a}[/math] is the value that maximizes or minimizes a quadratic function: Let [math]f(x) = ax^2+bx+c[/math]. Students will sketch the graphs of quadratic functions using parameters of an equation. The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. 2 Determining Maximum and Minimum Values of Quadratic Functions Completing the Square The process of completing the square is used to convert quadratic functions in standard form to vertex form. Then find the minimum or maximum value. Unit 1- Function Families Quadratic Functions The graph of a quadratic function is called a _____. Example: Find the minimum or maximum value of f(x) = –3 x2 + 2. Students interpret quadratic functions from graphs and tables: zeros ( T‐intercepts), U‐intercept, the minimum or maximum value (vertex), the graph’s axis of symmetry, positive and negative values for the function,. [MODULE 7:GRAPHS OF QUADRATIC FUNCTIONS] New York City College of Technology MAT 1275 PAL Workshops 1 Supported by CUNY OAA-Improving Undergraduate Learning Outcomes in Mathematics, BMI, NSF STEP #0622493,and Perkins. inverse function: the reciprocal function itself. Then find the minimum or maximum value and determine where it occurs. Our book does this with the use of graphing calculators, but I was. 1 Understand that a function from one set (the input, called the. y = a(x - h) 2 + k. Next, use the R function optimize() to find the value of that maximum. How do find the minimum/maximum point on a curve? and can you give an example please By the way keep this to Core 1 level/AS level Exam board is edexcel. If is negative, the maximum value of the function is. (This is a concept often used in solving real-world problems. To find \( m \), use the derivative of the function. Graphing Scatter Plots and Mathematical Models on the TI-83. We can help you solve an equation of the form "ax 2 + bx + c = 0" Just enter the values of a, b and c below:. Note that we did a Quadratic Inequality Real World Example here. Determine a quadratic function's minimum or maximum value There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue. Find a quadratic function with minimum value of -4 when x = 3 and one zero of 1 A rectangular dog kennel is to be constructed alongside a house with 60m of fencing. Quadratic functions are usually the first we encounter that have curved or nonlinear graphs. → Read More: Max/Min Problem – Maximizing Revenue, Selling Calculators. Graphing Quadratic Functions in Standard Form 7. C program to find maximum and minimum using functions February 19, 2016 Pankaj C programming C , Function , Program Write a C program to input two or more numbers from user and find maximum and minimum of the given numbers using functions. The Solver can find the maximum and minimum values, as well as the roots to an equation or a system of equations. Minimum and Maximum Values Let f(x) = ax2 + bx + c, where a = 0. Set the derivative equal to zero and. Given a quadratic function ax 2 + bx + c. A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0. Calculation of Maximum and Minimum Value 2 O ccun O Ccim ol b- ac aA. Here, we are interested in using scipy. Mathematics of finance, matrices, linear programming, and set operations. The graph of a quadratic function is a curve called a _____. Range of Quadratic Functions Substituting any real value of x into a quadratic equation results in a real number. Completing the square maxima and minima mc-TY-completingsquare1-2009-1 Completing the square is an algebraic technique which has several applications. This problem is equivalent to that of maximizing a polynomial, since any maximum of a quadratic polynomial p occurs at a minimum of the quadratic polynomial –p. f(x) = 7 - x^2 12. (The x value will be the time. The graphs of quadratic functions are more complex than linear and exponential because they include a turning point that is either the location of a maximum or a minimum. Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Functions Extreme Points Calculator calculus-function-extreme-points. The fact-checkers, whose work is more and more important for those who prefer facts over lies, police the line between fact and falsehood on a day-to-day basis, and do a great job. Today, my small contribution is to pass along a very good overview that reflects on one of Trump’s favorite overarching falsehoods. Namely: Trump describes an America in which everything was going down the tubes under  Obama, which is why we needed Trump to make America great again. And he claims that this project has come to fruition, with America setting records for prosperity under his leadership and guidance. “Obama bad; Trump good” is pretty much his analysis in all areas and measurement of U.S. activity, especially economically. Even if this were true, it would reflect poorly on Trump’s character, but it has the added problem of being false, a big lie made up of many small ones. Personally, I don’t assume that all economic measurements directly reflect the leadership of whoever occupies the Oval Office, nor am I smart enough to figure out what causes what in the economy. But the idea that presidents get the credit or the blame for the economy during their tenure is a political fact of life. Trump, in his adorable, immodest mendacity, not only claims credit for everything good that happens in the economy, but tells people, literally and specifically, that they have to vote for him even if they hate him, because without his guidance, their 401(k) accounts “will go down the tubes.” That would be offensive even if it were true, but it is utterly false. The stock market has been on a 10-year run of steady gains that began in 2009, the year Barack Obama was inaugurated. But why would anyone care about that? It’s only an unarguable, stubborn fact. Still, speaking of facts, there are so many measurements and indicators of how the economy is doing, that those not committed to an honest investigation can find evidence for whatever they want to believe. Trump and his most committed followers want to believe that everything was terrible under Barack Obama and great under Trump. That’s baloney. Anyone who believes that believes something false. And a series of charts and graphs published Monday in the Washington Post and explained by Economics Correspondent Heather Long provides the data that tells the tale. The details are complicated. Click through to the link above and you’ll learn much. But the overview is pretty simply this: The U.S. economy had a major meltdown in the last year of the George W. Bush presidency. Again, I’m not smart enough to know how much of this was Bush’s “fault.” But he had been in office for six years when the trouble started. So, if it’s ever reasonable to hold a president accountable for the performance of the economy, the timeline is bad for Bush. GDP growth went negative. Job growth fell sharply and then went negative. Median household income shrank. The Dow Jones Industrial Average dropped by more than 5,000 points! U.S. manufacturing output plunged, as did average home values, as did average hourly wages, as did measures of consumer confidence and most other indicators of economic health. (Backup for that is contained in the Post piece I linked to above.) Barack Obama inherited that mess of falling numbers, which continued during his first year in office, 2009, as he put in place policies designed to turn it around. By 2010, Obama’s second year, pretty much all of the negative numbers had turned positive. By the time Obama was up for reelection in 2012, all of them were headed in the right direction, which is certainly among the reasons voters gave him a second term by a solid (not landslide) margin. Basically, all of those good numbers continued throughout the second Obama term. The U.S. GDP, probably the single best measure of how the economy is doing, grew by 2.9 percent in 2015, which was Obama’s seventh year in office and was the best GDP growth number since before the crash of the late Bush years. GDP growth slowed to 1.6 percent in 2016, which may have been among the indicators that supported Trump’s campaign-year argument that everything was going to hell and only he could fix it. During the first year of Trump, GDP growth grew to 2.4 percent, which is decent but not great and anyway, a reasonable person would acknowledge that — to the degree that economic performance is to the credit or blame of the president — the performance in the first year of a new president is a mixture of the old and new policies. In Trump’s second year, 2018, the GDP grew 2.9 percent, equaling Obama’s best year, and so far in 2019, the growth rate has fallen to 2.1 percent, a mediocre number and a decline for which Trump presumably accepts no responsibility and blames either Nancy Pelosi, Ilhan Omar or, if he can swing it, Barack Obama. I suppose it’s natural for a president to want to take credit for everything good that happens on his (or someday her) watch, but not the blame for anything bad. Trump is more blatant about this than most. If we judge by his bad but remarkably steady approval ratings (today, according to the average maintained by 538.com, it’s 41.9 approval/ 53.7 disapproval) the pretty-good economy is not winning him new supporters, nor is his constant exaggeration of his accomplishments costing him many old ones). I already offered it above, but the full Washington Post workup of these numbers, and commentary/explanation by economics correspondent Heather Long, are here. On a related matter, if you care about what used to be called fiscal conservatism, which is the belief that federal debt and deficit matter, here’s a New York Times analysis, based on Congressional Budget Office data, suggesting that the annual budget deficit (that’s the amount the government borrows every year reflecting that amount by which federal spending exceeds revenues) which fell steadily during the Obama years, from a peak of $1.4 trillion at the beginning of the Obama administration, to $585 billion in 2016 (Obama’s last year in office), will be back up to $960 billion this fiscal year, and back over $1 trillion in 2020. (Here’s the New York Times piece detailing those numbers.) Trump is currently floating various tax cuts for the rich and the poor that will presumably worsen those projections, if passed. As the Times piece reported: