Grade 8 Algebra I Characteristics of Quadratic Functions CONFIDENTIAL

Tell whether each function is quadratic. Explain. Warm Up Tell whether each function is quadratic. Explain. 1) y + x = 2x2 2) y = -3x + 20 3) (-2, 4) , (-1, 1) , (0, 0) , (1, 1) , (2, 4) CONFIDENTIAL

Characteristics of Quadratic Functions An x-intercept of a function is a value of x when y = 0. A zero of a function is an x-value that makes the function equal to 0. So a zero of a function is the same as an x-intercept of a function. Since a graph intersects the x-axis at the point or points containing an x-intercept these intersections are also at the zeros of the function. A quadratic function may have one, two, or no zeros. CONFIDENTIAL

Finding Zeros of Quadratic Functions From Graphs Find the zeros of each quadratic function from its graph. Check your answer. A) y = x2 - x - 2 -2 2 x y 4 The zeros appear to be -1 and 2. Check: y = x2 - x – 2 y = (-1)2 – (-1) - 2 = 1 + 1 -2 = 0 y = (2)2 – (2) – 2 = 4 - 2 - 2 = 0 CONFIDENTIAL

The only zero appears to be 1. B) y= -2x2 + 4x - 2 The only zero appears to be 1. -2 2 x y -4 Check: y = -2x2 + 4x - 2 y = -2(1)2 + 4(1) - 2 = -2 + 4 - 2 = 0 CONFIDENTIAL

C) y = x2 + 1 4 -2 2 x y 4 The graph does not cross the x-axis, so there are no zeros of this function. CONFIDENTIAL

Now you try! Find the zeros of each quadratic function from its graph. Check your answer. 1) y=-4x2 - 2 2) y= x2 - 6x + 9 -2 2 x y -4 -6 y 4 2 x 6 CONFIDENTIAL

Finding the Axis of Symmetry by Using Zeros A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry. The axis of symmetry always passes through the vertex of the parabola. You can use the zeros to find the axis of symmetry. One Zero If a function has one zero, use the x-coordinate of the vertex to find the axis of symmetry. y 4 2 x 6 Vertex: (3, 0) Axis of symmetry: x = 3 CONFIDENTIAL

If a function has two zeros, use the average of the two zeros to find the axis of symmetry. -2 2 x y -4 -6 4 -4, 0 0, 0 x = -2 -4 + 0 = -4 = -2 2 2 Axis of symmetry: x = -2 CONFIDENTIAL

Finding the Axis of Symmetry by Using Zeros Find the axis of symmetry of each parabola. -2 2 x y -4 -6 4 A) Identify the x-coordinate of the vertex. (2, 0) The axis of symmetry is x = 2. CONFIDENTIAL

Find the average of the zeros. y B) 1 + 5 = 6 = 3 2 2 2 x Find the average of the zeros. -2 2 4 6 -4 -2 -4 -6 -8 The axis of symmetry is x = 3. CONFIDENTIAL

Find the axis of symmetry of each parabola. Now you try! Find the axis of symmetry of each parabola. A) B) CONFIDENTIAL

Finding the Axis of Symmetry by Using the Formula If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry. The formula works for all quadratic functions. FORMULA For a quadratic function y = ax2 + bx + c, the axis of symmetry is the vertical line x = - b 2a EXAMPLE y = 2x2 + 4x + 5 x = - b 2a x = - 4 = -1 2(2) The axis of symmetry is x = -1. CONFIDENTIAL

Finding the Axis of Symmetry by Using the Formula Find the axis of symmetry of the graph of y = x2 + 3x + 4. Step1 Find the values of a and b. y = 1x2 + 3x + 4 a = 1, b = 3 Step2 Use the formula x = - b 2a x = - 3 . = - 3 = -1.5 2(1) 2 The axis of symmetry is x = -1.5. CONFIDENTIAL

1) Find the axis of symmetry of the graph of y = 2x2 + x + 3. Now you try! 1) Find the axis of symmetry of the graph of y = 2x2 + x + 3. CONFIDENTIAL

Finding the Vertex of a Parabola Once you have found the axis of symmetry, you can use it to identify the vertex. Step 1: To find the x-coordinate of the vertex, find the axis of symmetry by using zeros or the formula. Step 2: To find the corresponding y-coordinate, substitute the x-coordinate of the vertex into the function. Step 3: Write the vertex as an ordered pair. CONFIDENTIAL

Finding the Vertex of a Parabola Find the vertex. A) y = - x2 - 2x Step 1: Find the x-coordinate. The zeros are -2 and 0. X = -2 + 0 = -2 = -1 2 2 Step 2: Find the corresponding y-coordinate. y = - x2 - 2x Use the function rule. = - (-1)2 - 2(-1) = 1 Substitute -1 for x. Step 3: Write the ordered pair. (-1, 1) The vertex is (-1, 1) . CONFIDENTIAL

Step 1: Find the x-coordinate. B) y = 5x2 - 10x + 3 Step 1: Find the x-coordinate. a = 5, b = -10 Identify a and b. x = b = - (- 10) = 10 = 1 2a 2(5) 10 Substitute 5 for a and -10 for b. Step 2: Find the corresponding y-coordinate. y = 5x2 - 10x + 3 Use the function rule. = 5(1)2 - 10 (1) + 3 = 5 - 10 + 3 = -2 Substitute 1 for x. Step 3: Write the ordered pair. (1, -2) The vertex is (1, -2) . CONFIDENTIAL

1) Find the vertex of the graph of y = x2 - 4x - 10. Now you try! 1) Find the vertex of the graph of y = x2 - 4x - 10. CONFIDENTIAL

Architecture Application The height above water level of a curved arch support for a bridge can be modeled by f (x) = -0.007x2 + 0.84x + 0.8, where x is the distance in feet from where the arch support enters the water. Can a sailboat that is 24 feet tall pass under the bridge? Explain. The vertex represents the highest point of the arch support. Step 1: Find the x-coordinate. a = -0.007, b = 0.84 Identify a and b. x = -b = - (0.84) = 60 2a 2(-0.007) Substitute -0.007 for a and 0.84 for b. CONFIDENTIAL Next page 

Step 2: Find the corresponding y-coordinate. f (x) = -0.007x2 + 0.84x + 0.8 Identify a and b. = -0.007(60)2 + 0.84(60) + 0.8 = 26 Substitute 60 for x. Since the height of the arch support is 26 feet, the sailboat can pass under the bridge. CONFIDENTIAL

Now you try! 1) The height of a small rise in a roller coaster track is modeled by f (x) = -0.07x2 + 0.42x + 6.37, where x is the distance in feet from a support pole at ground level. Find the height of the rise. CONFIDENTIAL

Find the zeros of each quadratic function from its graph. Assessment Find the zeros of each quadratic function from its graph. 2) 1) CONFIDENTIAL

Find the axis of symmetry of each parabola. 3) 4) CONFIDENTIAL

For each quadratic function, find the axis of symmetry of its graph. 5) y = x2 + 4x - 7 6) y = 3x2 - 18x + 1 7) y = 2x2 + 3x - 4 CONFIDENTIAL

Find the vertex of each parabola. 8) y = -5x2 + 10x + 3 9) y = x2 + 4x - 7 10) The height in feet above the ground of an arrow after it is shot can be modeled by y = -16t2 + 63t + 4. Can the arrow pass over a tree that is 68 feet tall? Explain. CONFIDENTIAL

Characteristics of Quadratic Functions Let’s review Characteristics of Quadratic Functions An x-intercept of a function is a value of x when y = 0. A zero of a function is an x-value that makes the function equal to 0. So a zero of a function is the same as an x-intercept of a function. Since a graph intersects the x-axis at the point or points containing an x-intercept these intersections are also at the zeros of the function. A quadratic function may have one, two, or no zeros. CONFIDENTIAL

Finding Zeros of Quadratic Functions From Graphs Find the zeros of each quadratic function from its graph. Check your answer. A) y = x2 - x - 2 -2 2 x y 4 The zeros appear to be -1 and 2. Check: y = x2 - x – 2 y = (-1)2 – (-1) - 2 = 1 + 1 -2 = 0 y = (2)2 – (2) – 2 = 4 - 2 - 2 = 0 CONFIDENTIAL

C) y = x2 + 1 4 -2 2 x y 4 The graph does not cross the x-axis, so there are no zeros of this function. CONFIDENTIAL

Finding the Axis of Symmetry by Using Zeros A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry. The axis of symmetry always passes through the vertex of the parabola. You can use the zeros to find the axis of symmetry. One Zero If a function has one zero, use the x-coordinate of the vertex to find the axis of symmetry. y 4 2 x 6 Vertex: (3, 0) Axis of symmetry: x = 3 CONFIDENTIAL

If a function has two zeros, use the average of the two zeros to find the axis of symmetry. -2 2 x y -4 -6 4 -4, 0 0, 0 x = -2 -4 + 0 = -4 = -2 2 2 Axis of symmetry: x = -2 CONFIDENTIAL

Finding the Axis of Symmetry by Using the Formula If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry. The formula works for all quadratic functions. FORMULA For a quadratic function y = ax2 + bx + c, the axis of symmetry is the vertical line x = - b 2a EXAMPLE y = 2x2 + 4x + 5 x = - b 2a x = - 4 = -1 2(2) The axis of symmetry is x = -1. CONFIDENTIAL

Finding the Axis of Symmetry by Using the Formula Find the axis of symmetry of the graph of y = x2 + 3x + 4. Step1 Find the values of a and b. y = 1x2 + 3x + 4 a = 1, b = 3 Step2 Use the formula x = - b 2a x = - 3 . = - 3 = -1.5 2(1) 2 The axis of symmetry is x = -1.5. CONFIDENTIAL

You did a great job today! CONFIDENTIAL