Lesson 9.2: Graph Essential Question: How do you graph general quadratic functions? Common Core CC.9-12.F.BF.3 Graph linear and quadratic functions and show intercepts, maxima, and minima. Warm-up:

9-2 Graph General Quadratic Functions Homework: Part One: 638/3-11; Part Two: 638/ even; 40-41; even

a>0a<0 a = 7 b = 2 c = 11 The graph opens upward. a = -3 b = 9 c = 4 The graph opens downward.

Does the vertex contain a minimum or a maximum value? a>0 y value in vertex is a MINIMUM a<0 y value in vertex is a MAXIMUM Tell whether the graph has a maximum value or minimum value.

Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Upward. Minimum. Upward. Minimum. Downward. Maximum. Downward. Maximum. Upward. Minimum. Upward. Minimum.

Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Upward. Minimum. Upward. Minimum. Downward. Maximum. Downward. Maximum. Upward. Minimum. Upward. Minimum.

Axis of symmetry is the line: The x-coordinate of the vertex is on this line. To find the y-coordinate of the vertex, substitute the value of x into the original equation.

Vertex: Axis of Symmetry: This is the Minimum.

Vertex: Axis of Symmetry: This is the Maximum.

downup > < c 0 c

Graph y = -2x 2 + 4x + 1 Step 1: Find the axis of symmetry. x = 1 Step 2: Find the y-coordinate of the vertex. Substitute the value of x into the original equation. y = -2(1) 2 + 4(1) + 1 y = 3 Step 3: Find the y-intercept. Substitute 0 for x in the original equation. y = -2(0) 2 + 4(0) + 1 y = 1 Step 4: Choose another value for x on the same side of the vertex. Substitute -1 for x in the original equation. y = -2(-1) 2 + 4(-1) + 1 y = -5 Step 5: Reflect the points across the axis of symmetry and draw the parabola.

Graph y = x 2 – 6x + 9 Step One: Find the axis of symmetry. Step Two: Find the y-coordinate of the vertex. Substitute the value of x into the original equation. Step Three: Find the y-intercept. Substitute 0 for x in the original equation. Step Four: Choose another value for x on the same side of the vertex. Substitute for x in the original equation. Step Five: Reflect the points across the axis and draw the parabola. y x

This function has a minimum. The minimum value is y = 3.

Upward. Minimum. Upward. Minimum. Downward. Maximum. Downward. Maximum. Upward. Minimum. Upward. Minimum.

EXAMPLE 3 Find the minimum or maximum value Tell whether the function f ( x ) = – 3x 2 – 12x + 10 has a minimum value or a maximum value. Then find the minimum or maximum value. SOLUTION Because a = – 3 and – 3 < 0, the parabola opens down and the function has a maximum value. To find the maximum value, find the vertex. x = – = – = – 2 b 2a2a – 12 2(– 3) f(–2) = – 3(–2) 2 – 12(–2) + 10 = 22 Substitute –2 for x. Then simplify. The x- coordinate is – b 2a2a

Find the minimum or maximum value EXAMPLE 3 ANSWER The maximum value of the function is f ( – 2 ) = 22.

Find the minimum value of a function EXAMPLE 4 The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled by the graph of y = x 2 – 0.37x where x and y are measured in feet. What is the height of the cable above the water at its lowest point? SUSPENSION BRIDGES

Find the minimum value of a function EXAMPLE 4 SOLUTION The lowest point of the cable is at the vertex of the parabola. Find the x- coordinate of the vertex. Use a = and b = – x = – = – ≈ 1910 b 2a2a – ( ) Use a calculator. Substitute 1910 for x in the equation to find the y -coordinate of the vertex. y ≈ (1910) 2 – 0.37(1910) ≈ 196

EXAMPLE 4 ANSWER The cable is about 196 feet above the water at its lowest point. Find the minimum value of a function

GUIDED PRACTICE for Examples 3 and 4 3. Tell whether the function f ( x ) = 6x x + 13 has a minimum value or a maximum value. Then find the minimum or maximum value.  1 2 Minimum value; ANSWER

GUIDED PRACTICE for Examples 3 and 4 SUSPENSION BRIDGES 4. The cables between the two towers of the Takoma Narrows Bridge form a parabola that can be modeled by the graph of the equation y = x 2 – 0.4x where x and y are measured in feet. What is the height of the cable above the water at its lowest point ? Round your answer to the nearest foot. ANSWER 221 feet

The highest point occurs at the Maximum. The y value of the vertex is 10. The height of the dome is 10 feet. The highest point occurs at the Maximum. The y value of the vertex is 10. The height of the dome is 10 feet.

Summary

Classwork/Homework 9.2 Practice B worksheet