Alg1 Notes 8.2.notebook
March 25, 2013
8.2 Characteristics of Quadratic Functions
1. Is y = –x – 1 quadratic? Explain. Use this graph for Problems 3­5.
2. Graph y = 1.5x2. 3. Identify the vertex
4. Does the function have a minimum or maximum? What is it? 5. Find the domain and range.
Find the x­intercept of each linear function.
7. y = ­2/3x + 7/3
8. y = 3x + 6 6. y = 2x – 3
Evaluate each quadratic function for the given input values.
9. y = –3x2 + x – 2, when x = 2 10. y = x2 + 2x + 3, when x = –1
8.2 Warm Up Answers
1. Is y = –x – 1 quadratic? Explain. Use this graph for Problems 3­5.
2. Graph y = 1.5x2. 3. Identify the vertex
4. Does the function have a minimum or maximum? What is it? 5. Find the domain and range.
Stop here for Monday
Find the x­intercept of each linear function.
7. y = ­2/3x + 7/3
8. y = 3x + 6 6. y = 2x – 3
Evaluate each quadratic function for the given input values.
9. y = –3x2 + x – 2, when x = 2 10. y = x2 + 2x + 3, when x = –1
Alg1 Notes 8.2.notebook
March 25, 2013
8.2 Characteristics of Quadratic Functions
Learning Targets:
1. Find the zeros of a quadratic function from its graph.
2. Find the axis of symmetry and the vertex of a parabola. How we use this...
Engineers can use characteristics
of quadratic functions to find the
height of the arch supports of
bridges.
I. Identifying Zeros of Quadratic Functions from Graphs
A zero of a function is an x­value that makes the function equal to 0. Graphically it is the x­intercept.
Find the zeros of the quadratic function from its graph. Check your answer.
1. y = x2 – 2x – 3 The zeros appear to be ___ and ___. Check..
y = x2 – 2x – 3 Alg1 Notes 8.2.notebook
March 25, 2013
I. Identifying Zeros of Quadratic Functions from Graphs
Find the zeros of the quadratic function from its graph. Check your answer.
2. y = x2 + 8x + 16 The zero appears to be ___. Check..
I. Identifying Zeros of Quadratic Functions from Graphs
Find the zeros of the quadratic function from its graph. Check your answer.
3. y = –2x2 – 2 The graph does not cross the x­axis, so there are no zeros of this function.
Alg1 Notes 8.2.notebook
March 25, 2013
Find the zeros of the quadratic function from its graph. Check your answer.
4. y = –4x2 – 2 5. y = x2 – 6x + 9 II. Axis of Symmetry of Quadratic Functions
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.
Alg1 Notes 8.2.notebook
II. Finding Axis of Symmetry Using Zeros
Find the axis of symmetry of each parabola.
6. Find the axis of symmetry of each parabola.
7. March 25, 2013
Alg1 Notes 8.2.notebook
March 25, 2013
III. Axis of Symmetry Using 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.
*8. Find the axis of symmetry of y = –3x2 + 10x + 9.
9. Find the axis of symmetry of y = 2x2 + x + 3.
IV. Finding the Vertex of a Parabola
Once you have found the axis of symmetry, you can use it to identify the vertex.
For Example, let's use the last problem: y = 2x2 + x + 3
Alg1 Notes 8.2.notebook
March 25, 2013
IV. Finding the Vertex of a Parabola
Find the vertex.
10. y = 0.25x2 + 2x + 3
Step 1 Find the x­coordinate of the vertex. The zeros are –6 and –2. Step 2 Find the corresponding y­coordinate
Step 3 Write the ordered pair.
The vertex is (–4, –1).
Find the vertex.
11. y = –3x2 + 6x – 7 12. y = x2 – 4x – 10
Step 1 Find the x­coordinate of the vertex. Step 2 Find the corresponding y­coordinate
Step 3 Write the ordered pair.
The vertex is (1, –4).
The vertex is (2, –14).
Alg1 Notes 8.2.notebook
March 25, 2013
V. Application
The graph of f(x) = –0.06x2 + 0.6x + 10.26 can be used to model the height in meters of an arch support for a bridge, where the x­axis represents the water level and x represents the horizontal distance in meters from where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? Explain.
Hint: The vertex represents the highest point of the arch support. Work with your partner and provide a viable argument for your answer.
Since the height of each support is 11.76 m, the sailboat cannot pass under the bridge.
8.2 Page 535 #3 ­ 18, 19­33 odd, 35, 41 ­ 45