Chapter 2 Functions and Graphs Section 3 Quadratic Functions

Learning Objectives for Section 2.3 Quadratic Functions The student will be able to identify and define quadratic functions, equations, and inequalities. The student will be able to identify and use properties of quadratic functions and their graphs. The student will be able to solve applications of quadratic functions. Barnett/Ziegler/Byleen Business Calculus 12e

Quadratic Functions If a, b, c are real numbers with a not equal to zero, then the function is a quadratic function and its graph is a parabola. Barnett/Ziegler/Byleen Business Calculus 12e

Vertex Form of the Quadratic Function It is convenient to convert the general form of a quadratic equation to what is known as the vertex form: This is done by completing the square which will be reviewed in a few slides. Barnett/Ziegler/Byleen Business Calculus 12e

Generalization For Vertex is at (h , k) If a > 0, the graph opens upward. If a < 0, the graph opens downward. Axis of symmetry: x = h k is the minimum if a > 0, otherwise its the maximum Domain: −∞, ∞ Range: If a < 0  −∞, 𝑘 If a > 0  𝑘, ∞ Barnett/Ziegler/Byleen Business Calculus 12e

Generalization Barnett/Ziegler/Byleen Business Calculus 12e

General Form to Vertex Form Completing the Square The example below illustrates the procedure: Consider Complete the square to find the vertex. f (x) = (3x2 – 6x) –1 Group first two terms f (x) = 3(x2 – 2x) –1 Factor out coef. of x2 f (x) = 3(x2 – 2x +1) –1 – 3 Complete the square inside the parentheses Since you’re really adding 3, you have to subtract 3 f (x) = 3(x – 1)2 – 4 Vertex (1, -4); opens upwards Axis of sym: x = 1; Minimum = -4 D: (-,  ); R: [-4, ) 𝑓 𝑥 = 3𝑥 2 −6𝑥−1 Barnett/Ziegler/Byleen Business Calculus 12e

Example Rewrite the function in vertex form: 𝑓 𝑥 =− 𝑥 2 +8𝑥−9 𝑓 𝑥 =(− 𝑥 2 +8𝑥)−9 𝑓 𝑥 =−( 𝑥 2 −8𝑥)−9 𝑓 𝑥 =− 𝑥 2 −8𝑥+16 −9+16 𝑓 𝑥 =− 𝑥−4 2 +7 Vertex (4, 7); opens downwards Axis of sym: x = 4; Maximum = 7 D: (-,  ); R: (-, 7] Barnett/Ziegler/Byleen Business Calculus 12e

Intercepts Y-intercept Plug in x = 0 Barnett/Ziegler/Byleen Business Calculus 12e

Intercepts y−intercept is:−1 Find the y intercept of: Barnett/Ziegler/Byleen Business Calculus 12e

Intercepts X-intercepts It might have 0, 1, or 2 x-intercepts They can be determined by: Factoring (if possible) Completing the square Quadratic Formula Barnett/Ziegler/Byleen Business Calculus 12e

Intercepts Find the x intercepts of 𝑓 𝑥 = 𝑥 2 +5𝑥−14 0=(𝑥+7)(𝑥−2) 𝑥+7=0 𝑥−2=0 𝑥=−7 𝑥=2 The x−intercepts are:−7 and 2. Barnett/Ziegler/Byleen Business Calculus 12e

Intercepts Find the x intercepts of Using the quadratic formula: 𝑥= −6± 6 2 −4(−3)(−1) 2(−3) = −6± 24 −6 Exact Approx. = −6±2 6 −6 = −3± 6 −3 ≈1.82, 0.18 Barnett/Ziegler/Byleen Business Calculus 12e

Finding x-intercepts Using a Graphing Calculator Graph: y = –x2 + 5x + 3 Select CALC (2nd Trace) Select 2: zero Left bound? Use arrows to position cursor to the left of intercept, then hit ENTER. Right bound? Use arrows to position cursor to the right of intercept, then hit ENTER. Guess? Hit ENTER. Zero  -0.5414 Repeat to find other zero. Zero  5.5414 Barnett/Ziegler/Byleen Business Calculus 12e

Max and Min Values A parabola that opens upwards has a minimum value. A parabola that opens downwards has a maximum value. In either case, the max/min value is the y-coordinate of the vertex. Finding the vertex from the equation: 𝑓 𝑥 =𝑎 (𝑥−ℎ) 2 +𝑘  Vertex (h, k) 𝑓 𝑥 = 𝑎𝑥 2 +𝑏𝑥+𝑐  Vertex −𝑏 2𝑎 , 𝑓 −𝑏 2𝑎 Barnett/Ziegler/Byleen Business Calculus 12e

Example Find the maximum or minimum of each function: 𝑓 𝑥 = −(𝑥+3) 2 +7 𝑓 𝑥 = 3𝑥 2 −12𝑥+14 − 𝑏 2𝑎 =− −12 6 =2 𝑉𝑒𝑟𝑡𝑒𝑥 (−3, 7) 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑜𝑓 7. 𝑓 2 =2 𝑉𝑒𝑟𝑡𝑒𝑥 (2, 2) 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑜𝑓 2 Barnett/Ziegler/Byleen Business Calculus 12e

Finding Max/Min Using Graphing Calculator Graph: y = –x2 + 5x + 3 Select CALC (2nd Trace) Select 4: maximum Left bound? Right bound? Guess? Maximum 9.25 Vertex: (2.5, 9.25) Barnett/Ziegler/Byleen Business Calculus 12e

Quadratic Inequalities Two Methods for Solving Quadratic Inequalities Algebraic Do not need to review this procedure yet. Graphing Calculator This is the procedure we will use for now. Barnett/Ziegler/Byleen Business Calculus 12e

Quadratic Inequalities Graph the function. Determine its x-intercepts using the Calc Zero function. If inequality is: f(x) > 0 then state the intervals for which the graph is above the x-axis. f(x) < 0 then state the intervals for which the graph is below the x-axis. f(x)  0 then state the intervals for which the graph is on or above the x-axis. f(x)  0 then state the intervals for which the graph is on or below the x-axis. Barnett/Ziegler/Byleen Business Calculus 12e

Solving Quadratic Inequalities Solve the quadratic inequality –x2 + 5x + 3 > 0 . x-intercepts are: -0.5414 and 5.5414 The graph is on or above the x-axis over the interval: [– 0.5414, 5.5414 ] Barnett/Ziegler/Byleen Business Calculus 12e

Solving Quadratic Inequalities Solve the quadratic inequality –x2 + 5x + 3 < 0 . x-intercepts are: -0.5414 and 5.5414 The graph is below the x-axis over the interval: (−∞, – 0.5414)  (5.5414 , ∞) Barnett/Ziegler/Byleen Business Calculus 12e

Applications There are many applications involving quadratic functions. Let’s look at an example… Barnett/Ziegler/Byleen Business Calculus 12e

Break-Even Analysis The financial department of a company that produces digital cameras has the revenue and cost functions for x million cameras as follows: R(x) = x(94.8 – 5x) C(x) = 156 + 19.7x. Both have domain 1 < x < 15 Break-even points are the production levels at which R(x) = C(x). Find the break-even points (using your graphing calculator) to the nearest million cameras. Barnett/Ziegler/Byleen Business Calculus 12e

Solution to Break-Even Problem (continued) If we graph the cost and revenue functions on a graphing utility, we obtain the following graphs, showing the two intersection points: x = 2.490 or 12.530 The company breaks even when they sell approximately 2 million cameras and 13 million cameras. Barnett/Ziegler/Byleen Business Calculus 12e

Quadratic Regression Insert speed in L1 and mpg in L2. Turn Plot 1 on Evinrude Outboard Motor Insert speed in L1 and mpg in L2. Turn Plot 1 on ZoomStat STAT CALC 5:QuadReg Regression equation is: 𝑦=−.00968 𝑥 2 +0.5069𝑥−0.1794 Speed (mph) Miles per Gallon 10.3 4.1 18.3 5.6 24.6 6.6 29.1 6.4 33.0 6.1 36.0 5.4 38.9 4.9 Barnett/Ziegler/Byleen Business Calculus 12e