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Algebra 1

7.11.3 Sketching a Graph of a Quadratic Function Using at Least Three Identifiable Points

Algebra 17.11.3 Sketching a Graph of a Quadratic Function Using at Least Three Identifiable Points

7.11.3 Sketching a Graph of a Quadratic Function Using at Least Three Identifiable Points

7.11.3 • Sketching a Graph of a Quadratic Function Using at Least Three Identifiable Points

Activity

1. The functions f f , g g , and h h are given. Refer to these functions to complete questions a–f.

Equation x x -intercept x x -coordinate of the vertex

f ( x ) = ( x + 3 ) ( x − 5 ) f ( x ) = ( x + 3 ) ( x − 5 )

a. _____

b. _____

g ( x ) = 2 x ( x − 3 ) g ( x ) = 2 x ( x − 3 )

c. _____

d. _____

h ( x ) = ( x + 4 ) ( 4 − x ) h ( x ) = ( x + 4 ) ( 4 − x )

e. _____

f. _____

a. Predict the x x -intercepts for function f ( x ) = ( x + 3 ) ( x − 5 ) f ( x ) = ( x + 3 ) ( x − 5 ) .

b. Predict the x x -coordinate of the vertex for function f ( x ) = ( x + 3 ) ( x − 5 ) f ( x ) = ( x + 3 ) ( x − 5 ) .

c. Predict the x x -intercepts for function g ( x ) = 2 x ( x − 3 ) g ( x ) = 2 x ( x − 3 ) .

d. Predict the x x -coordinate of the vertex for function g ( x ) = 2 x ( x − 3 ) g ( x ) = 2 x ( x − 3 ) .

e. Predict the x x -intercepts for function h ( x ) = ( x + 4 ) ( 4 − x ) h ( x ) = ( x + 4 ) ( 4 − x ) .

f. Predict the x x -coordinate of the vertex for function h ( x ) = ( x + 4 ) ( 4 − x ) h ( x ) = ( x + 4 ) ( 4 − x ) .

2. Use the graphing tool or technology outside the course. Graph the functions f f , g g , and h h :
 f ( x ) = ( x + 3 ) ( x − 5 ) f ( x ) = ( x + 3 ) ( x − 5 )
g ( x ) = 2 x ( x − 3 ) g ( x ) = 2 x ( x − 3 )
h ( x ) = ( x + 4 ) ( 4 − x ) h ( x ) = ( x + 4 ) ( 4 − x )

Use the graphs to check your predictions.

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3. Without using technology, sketch a graph that represents the equation y = ( x − 7 ) ( x + 11 ) y = ( x − 7 ) ( x + 11 ) and that shows the x x -intercepts and the vertex. Think about how to find the y y -coordinate of the vertex. Be prepared to explain your reasoning.

Are you ready for more?

Extending Your Thinking
1.

The quadratic function f f is given by f ( x ) = x 2 + 2 x + 6 f ( x ) = x 2 + 2 x + 6 .

Find f ( − 2 ) f ( − 2 ) .

2.

Find f ( 0 ) f ( 0 ) .

3.

What is the x x -coordinate of the vertex of the graph of this quadratic function?

4.

Does the graph have any x x -intercepts? Explain or show how you know.

Video: Sketching a Graph of a Quadratic Function 

Watch the following video to learn more about sketching a graph of a quadratic function using at least three identifiable points.

Sketching a Graph of a Quadratic Function

Self Check

Find the coordinate of the vertex of f ( x ) = ( x − 2 ) ( x − 4 ) .
  1. ( 0 , 8 )
  2. ( 4 , 0 )
  3. ( 3 , − 1 )
  4. ( 2 , 0 )

Additional Resources

Graphing Quadratics With Points

The function f f given by f ( x ) = ( x + 1 ) ( x − 1 ) f ( x ) = ( x + 1 ) ( x − 1 ) is written in factored form. Recall that this form is helpful for finding the zeros of the function (where the function has the value 0) and telling us the x x -intercepts on the graph representing the function.

Here is a graph representing f f . It shows two x x -intercepts at x = − 1 x = − 1 and x = 3 x = 3 .

If we use –1 and 3 as inputs to f f , what are the outputs?

  • f ( − 1 ) = ( − 1 + 1 ) ( − 1 − 3 ) = ( 0 ) ( − 4 ) = 0 f ( − 1 ) = ( − 1 + 1 ) ( − 1 − 3 ) = ( 0 ) ( − 4 ) = 0
  • f ( 3 ) = ( 3 + 1 ) ( 3 − 3 ) = ( 4 ) ( 0 ) = 0 f ( 3 ) = ( 3 + 1 ) ( 3 − 3 ) = ( 4 ) ( 0 ) = 0

Because the inputs –1 and 3 produce an output of 0, they are the zeros of the function f f . And because both x x values have 0 for their y y value, they also give us the x x -intercepts of the graph (the points where the graph crosses the x x -axis, which always have a y y -coordinate of 0). So, the zeros of a function have the same values as the x x -coordinates of the x x -intercepts of the graph of the function.

The factored form can also help us identify the vertex of the graph, which is the point where the function reaches its minimum value. Notice that the x x -coordinate of the vertex is 1, and that 1 is halfway between –1 and 3. Once we know the x x -coordinate of the vertex, we can find the y y -coordinate by evaluating the function f ( 1 ) = ( 1 + 1 ) ( 1 − 3 ) = ( 2 ) ( − 2 ) = − 4 f ( 1 ) = ( 1 + 1 ) ( 1 − 3 ) = ( 2 ) ( − 2 ) = − 4 . So the vertex is at ( 1 , − 4 ) ( 1 , − 4 ) .

When a quadratic function is in standard form, the y y -intercept is clear: its y y -coordinate is the constant term c c in a x 2 + b x + c a x 2 + b x + c . To find the y y -intercept from factored form, we can evaluate the function at x = 0 x = 0 , because the y y -intercept is the point where the graph has an input value of 0:

f ( 0 ) = ( 0 + 1 ) ( 0 − 3 ) = ( 1 ) ( − 3 ) = − 3 f ( 0 ) = ( 0 + 1 ) ( 0 − 3 ) = ( 1 ) ( − 3 ) = − 3

Try it

Graphing Quadratics with Points

Find the zeros and vertex of the function f f given by f ( x ) = ( x − 4 ) ( x + 2 ) f ( x ) = ( x − 4 ) ( x + 2 ) . Use the points to graph f f .

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