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Algebra 1

7.10.2 Quadratic Forms and Their Graphs

Algebra 17.10.2 Quadratic Forms and Their Graphs

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Activity

In an earlier lesson, we saw that an equation such as h ( t ) = 10 + 78 t 16 t 2 h ( t ) = 10 + 78 t 16 t 2 can model the height of an object thrown upward from a height of 10 feet with a vertical velocity of 78 feet per second.

A young man in a purple sweater stands on railroad tracks in a forest, looking up as he juggles or catches a red apple above his hands. Lush green trees frame the scene.

1. Is the expression 16 t 2 + 78 t + 10 16 t 2 + 78 t + 10 written in standard form? Be prepared to show your reasoning.

2. Paola said that the equation g ( t ) = 2 ( 8 t + 1 ) ( t 5 ) g ( t ) = 2 ( 8 t + 1 ) ( t 5 ) also defines the same function, written in factored form. Is Paola correct? Be prepared to show your reasoning.

3. Here is a graph representing both g ( t ) = 2 ( 8 t + 1 ) ( t 5 ) g ( t ) = 2 ( 8 t + 1 ) ( t 5 ) and h ( t ) = 10 + 78 t 16 t 2 h ( t ) = 10 + 78 t 16 t 2 .

A parabola on a coordinate grid. The x-axis scale is 1 and extends from 0 to 6. The y-axis scale is 10 extends from 0 to 140.

a. What is the y y -value of the y y -intercept?

b. What is the x x -value of the x x -intercept?

c. What do each of these points mean in this situation?

Self Check

The quadratic function y = ( x 2 ) ( x + 3 ) is graphed below.

What are all of the x -intercepts?

  1. x = 6
  2. x = 3 , x = 2
  3. x = 2 , x = 6
  4. x = 3

Additional Resources

Characteristics of Quadratic Graphs

The graph of y = ( x + 1 ) ( x 2 ) y = ( x + 1 ) ( x 2 ) is shown below.

What are the x x - and y y -intercepts?

A parabola on a coordinate grid. The x-axis scale is 5 and extends from approximately negative 10 to 10. The y-axis scale is 5 and extends from approximately negative 10 to 12.

The x x -intercepts are where the graph crosses the x x -axis.

This happens at ( 1 , 0 ) ( 1 , 0 ) and ( 2 , 0 ) ( 2 , 0 ) .

The y y -intercept is where the graph crosses the y y -axis.

This happens at ( 0 , 2 ) ( 0 , 2 ) .

Try it

Try It: Characteristics of Quadratic Graphs

The graph of y = ( x 4 ) ( x + 1 ) y = ( x 4 ) ( x + 1 ) is below.

What are the x x - and y y -intercepts of the graph?

A parabola on a coordinate grid. The x-axis scale is 5 and extends from approximately negative 10 to 10. The y-axis scale is 5 and extends from approximately negative 10 to 12.

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