Section 2.3. Learning Objectives. Graphing Quadratic Functions

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1 Section 2.3 Quadratic Functions Learning Objectives Quadratic function, equations, and inequities Properties of quadratic function and their graphs Applications More general functions Graphing Quadratic Functions The general technique for graphing quadratics is the same as for graphing linear equations. However, since quadratics graph as curvy lines (called parabolas ), rather than the straight lines generated by linear equations.

2 Quadratic Function A quadratic function, in mathematics, is a polynomial function of the form: The graph of a quadratic function is a parabola whose major axis is parallel to the y-axis. Definitions Every parabola has an axis of symmetry which is the line that runs down its center. This line divides the graph into two perfect halves. Axis of symmetry formula is x = - b/2a Definitions (Vertex) Parabolas always have a lowest point (or a highest point, if the parabola is upside-down). This point, where the parabola changes direction, is called the vertex. {- b/2a, f(-b/2a)} (Note: The a in the vertex form f(x) = a(x h) 2 + k of the quadratic is the same as the a in the common form of the quadratic equation, y = ax 2 + bx + c.)

3 Graphing Quadratic Functions positive quadratic y = x 2 negative quadratic y = x 2 Definitions (Vertex) Quadratic function in vertex form: f(x) = a(x h) 2 + k 1. Vertex is (h, k) 2. Axis of symmetry: x = h 3. y-intercept: Set x = 0 and solve for y. Or we can say, find f(0). 4. x-intercept: Set f(x) = 0 and solve for x. Solving a Quadratic Equation (Ex 1) Example 1 Quadratic Equation: y = x² + 2x + 1 where a = 1, b = 2, and c = 1 Using the quadratic formula to solve this equation just substitute a, b, and c into the general formula:

4 Quadratic Equation (Ex 1) Example 1 Quadratic Equation: y = x² + 2x + 1 where a = 1, b = 2, and c = 1 Solving a Quadratic Equation Example 1 Quadratic Equation: y = x² + 2x + 1 where a = 1, b = 2, and c = 1 Solving a Quadratic Equation (Ex 2) What is the range of y = x 2 4 x + 5 Step 1: Determine the min. values of the parabola Step 2: Use the formula: -b/2a => 4/2 = 2 Step 3: Substitute x into the equation y = 2 2 4(2) + 5 = 1 So the range is all points above the min. value or y 1

5 Solving a Quadratic Equation (Ex 2) Example 3 Find the x-intercepts and vertex of y = x 2 + 2x 4. To find the vertex, look at the coefficients: a = 1 and b = 2. Then: h = (2) / 2( 1) = 1 To find k, plug h in for x and simplify: k = (1) 2 + 2(1) 4 = = 2 5 = 3 (1, -3) Solution is the vertex Example 3 Now find some additional plot points using a T-chart:

6 Example 3 Pick x-values that were centered around the x-coordinate of the vertex and plot the parabola: Example 4 Solve the quadratic equation: x 2 = 32 4x 1. By factoring 2. With the quadratic equation 3. With your graphing calculator Example 4 Solve the quadratic equation: x 2 = 32 4x 1. By factoring (x 4)(x + 8) = 0 (4, - 8)

7 Example 4 Solve the quadratic equation: x 2 = 32 4x 1. With the quadratic equation Get a = 1; b = 4; and c =-32 Plug it in the equation and solve Example 4 Solve the quadratic equation: x 2 = 32 4x 1. With your graphing calculator Example 5 Given the function f(x) = x 2 4x + 2, complete the following parts. 1. State if the graph of f(x) opens upward or downward. 2. Find the vertex algebraically. Write answer in order pair. 3. State if the function has a maximum or min. Then give the max./min. value 4. Give the equation for the axis of symmetry. 5. Give the range of the function in interval notation.

8 Quadratic Function in Vertex Form (Ex 5) Example: Now sketch the graph of f(x) = x 2 4x + 2 Breakeven Analysis (Ex 6) The CFO of a company that makes Blackberry s has a revenue (in millions of dollars) and cost functions for x millions Blackberrys are given: R(x) = x(94.8 5x) C(x) = x Both have a domain 1 x 15 Use your graphing calculator to find the BE points to the nearest thousand Blackberrys. Breakeven Analysis (Ex 6) Graph the Profit function P(x) =

9 Quadratic Regression Example 7 A visual inspection of the plot of a data set might indicate that a parabola would be a better model of the data than a straight line. In that case, rather than using linear regression to fit a linear model to the data, we would use a quadratic regression on the graphing calculator to find the function of the form y = ax 2 + bx + c that best fits the data. Go to stat, choose 5. QuadReg Example of Quadratic Regression (Ex 7) An tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles). x Mileage Using the quadratic regression on the graphing calculator, find the quadratic function that best fits the date. Example of Quadratic Regression (Ex 7) An tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles). x Mileage Using the quadratic regression on the graphing calculator, find the quadratic function that best fits the date.

10 Example of Quadratic Regression (Ex 7) An tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles). What is the slope? What is the vertex? y = x x Graphing Quadratic Functions (Ex 8) The most basic quadratic is y = x 2. Graphing Quadratic Functions (Ex 8) The most basic quadratic is y = x 2.

11 More General Functions In mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x 2 4x + 7 is a polynomial, but x 2 4/x + 7x 3/2 is not, because its second term involves division by the variable x and also because its third term contains an exponent that is not a whole number. Graphs of Even Degree Polynomials with a positive leading coefficient with a negative leading coefficient Graphs of Odd Degree Polynomials with a positive leading coefficient with a negative leading coefficient

12 Why, why, why? Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions.