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Algebra 1

7.11.2 The Factored Form of a Quadratic Expression and the x-intercepts of Its Graph

Algebra 17.11.2 The Factored Form of a Quadratic Expression and the x-intercepts of Its Graph

7.11.2 The Factored Form of a Quadratic Expression and the x-intercepts of Its Graph

7.11.2 • The Factored Form of a Quadratic Expression and the x-intercepts of Its Graph

Activity

Consider two functions defined by f ( x ) = x ( x + 4 ) f ( x ) = x ( x + 4 ) and g ( x ) = x ( x − 4 ) g ( x ) = x ( x − 4 )

  1. Consider the function defined by f ( x ) = x ( x + 4 ) f ( x ) = x ( x + 4 ) .

Then, answer questions a–g, and create a table of values in which to enter the given x values and their outputs using the function.

a. When the value of x = − 4 x = − 4 , what is the value of f ( x ) f ( x ) ?

b. When the value of x = − 3 x = − 3 , what is the value of f ( x ) f ( x ) ?

c. When the value of x = 0 x = 0 , what is the value of f ( x ) f ( x ) ?

d. When the value of x = 1 x = 1 , what is the value of f ( x ) f ( x ) ?

e. When the value of x = 2 x = 2 , what is the value of f ( x ) f ( x ) ?

f. When the value of x = 3 x = 3 , what is the value of f ( x ) f ( x ) ?

g. When the value of x = 5 x = 5 , what is the value of f ( x ) f ( x ) ?

2. Now, answer questions a–d to analyze the function defined by f ( x ) = x ( x + 4 ) f ( x ) = x ( x + 4 ) .

a. What are the x x -intercepts of the graph? Be prepared to show your reasoning.

b. What is the value of the x x -coordinate in the vertex of the graph?

c. What is the value of the y y -coordinate in the vertex of the graph?

d. What is the vertex of the graph of function f f ? Be prepared to show your reasoning.

3. Consider the function defined by g ( x ) = x ( x − 4 ) g ( x ) = x ( x − 4 ) . Then, answer questions a–g, and create a table of values in which to enter the given x values and their outputs using the function.

a. When the value of x = − 4 x = − 4 , what is the value of g ( x ) g ( x ) ?

b. When the value of x = − 3 x = − 3 , what is the value of g ( x ) g ( x ) ?

c. When the value of x = 0 x = 0 , what is the value of g ( x ) g ( x ) ?

d. When the value of x = 1 x = 1 , what is the value of g ( x ) g ( x ) ?

e. When the value of x = 2 x = 2 , what is the value of g ( x ) g ( x ) ?

f. When the value of x = 4 x = 4 , what is the value of g ( x ) g ( x ) ?

g. When the value of x = 5 x = 5 , what is the value of g ( x ) g ( x ) ?

4. Now, answer questions a–d to analyze the function defined by g ( x ) = x ( x − 4 ) g ( x ) = x ( x − 4 ) .

a. What are the x x -intercepts of the graph? Be prepared to show your reasoning.

b. What is the value of the x x -coordinate in the vertex of the graph?

c. What is the value of the y y -coordinate in the vertex of the graph?

d. What is the vertex of the graph of function g g ? Be prepared to show your reasoning.

5. Use the graphing tool or technology outside the course.

Access multimedia content

a. Plot the points from the tables on the same coordinate plane. (Consider using different colors or markings for each set of points so you can tell them apart.)

b. Make a couple of observations about how the two graphs compare.

Self Check

The table for function f is below. What is the vertex of the graph of f ?

x -3 -2 -1 0 1 2
f ( x ) 0 -1 -3 -1 0 2
  1. ( 1 , 0 )
  2. ( − 1 , − 3 )
  3. ( 0 , − 1 )
  4. ( − 3 , 0 )

Additional Resources

Determining Parts of a Graph from a Table

Identify the x x -intercepts and vertex of the quadratic from the table:

The x x -intercepts are where the y y -values are 0.

This occurs at ( − 3 , 0 ) ( − 3 , 0 ) and ( 0 , 0 ) ( 0 , 0 ) .

The vertex occurs where the function will create a mirror image, or be symmetrical for all points. This often occurs halfway between the two zeros (when there are two zeros).

For this function, the vertex is at ( − 1 , − 2 ) ( − 1 , − 2 ) .

Try it

Try It: Determining Parts of a Graph from a Table

Determine the vertex from the table:

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