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Quadratic Equations and Graphs

7.16.3 Quadratic Equations and Graphs

7.16.3 • Quadratic Equations and Graphs

Activity

A series of graphs are given on different coordinate places. Graph A is a parabola that opens up with a vertex at (4, 1). Graph B is a parabola that opens up with a vertex at (negative 1, negative 4). Graph C is a parabola that opens down with a vertex at (4, 1). Graph D is a parabola that opens down with a vertex at (negative 1, negative 4). Graph E is a parabola that opens up with a vertex at (negative 4, negative 1). Graph F is a parabola that opens up with a vertex at (1, 4). On all graphs, the x-axis extends from negative 8 to 8 with a scale of 1. The y-axis extends from negative 14 to 14 with a scale of 2.

Your teacher will give you a set of cards. Each card contains an equation or a graph that represents a quadratic function. Take turns matching each equation to a graph that represents the same function.

  • For each pair of cards that you match, explain to your partner how you know they belong together.
  • For each pair of cards that your partner matches, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
  • Once all the cards are matched, record the equation, the letter of the equation, and a sketch of the corresponding graph, and write a brief note or explanation about how you knew they were a match.
1.

Choose the equation that matches this graph.

A clear graph of an orange parabola on a grid, illustrating a U-shaped curve. The parabola opens upwards, with its vertex located around the point (1, 4) and crossing the y-axis at (0, 5).

  1. f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4

  2. g ( x ) = − ( x − 4 ) 2 + 1 g ( x ) = − ( x − 4 ) 2 + 1

  3. h ( x ) = ( x + 1 ) 2 − 4 h ( x ) = ( x + 1 ) 2 − 4

  4. p ( x ) = − ( x + 1 ) 2 − 4 p ( x ) = − ( x + 1 ) 2 − 4

  5. q ( x ) = 2 ( x − 4 ) 2 + 1 q ( x ) = 2 ( x − 4 ) 2 + 1

  6. r ( x ) = ( x + 4 ) 2 − 1 r ( x ) = ( x + 4 ) 2 − 1

f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4

2.

Choose the equation that matches this graph.

...

  1. f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4

  2. g ( x ) = − ( x − 4 ) 2 + 1 g ( x ) = − ( x − 4 ) 2 + 1

  3. h ( x ) = ( x + 1 ) 2 − 4 h ( x ) = ( x + 1 ) 2 − 4

  4. p ( x ) = − ( x + 1 ) 2 − 4 p ( x ) = − ( x + 1 ) 2 − 4

  5. q ( x ) = 2 ( x − 4 ) 2 + 1 q ( x ) = 2 ( x − 4 ) 2 + 1

  6. r ( x ) = ( x + 4 ) 2 − 1 r ( x ) = ( x + 4 ) 2 − 1

g ( x ) = − ( x − 4 ) 2 + 1 g ( x ) = − ( x − 4 ) 2 + 1

3.

Choose the equation that matches this graph.

...

  1. f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4

  2. g ( x ) = − ( x − 4 ) 2 + 1 g ( x ) = − ( x − 4 ) 2 + 1

  3. h ( x ) = ( x + 1 ) 2 − 4 h ( x ) = ( x + 1 ) 2 − 4

  4. p ( x ) = − ( x + 1 ) 2 − 4 p ( x ) = − ( x + 1 ) 2 − 4

  5. q ( x ) = 2 ( x − 4 ) 2 + 1 q ( x ) = 2 ( x − 4 ) 2 + 1

  6. r ( x ) = ( x + 4 ) 2 − 1 r ( x ) = ( x + 4 ) 2 − 1

h ( x ) = ( x + 1 ) 2 − 4 h ( x ) = ( x + 1 ) 2 − 4

4.

Choose the equation that matches this graph.

A green parabola opens downwards, with its vertex in the second quadrant. The graph is plotted on a white grid with black x and y axes. The x-axis is labeled from -5 to 10, and the y-axis from -10 to 5.

  1. f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4

  2. g ( x ) = − ( x − 4 ) 2 + 1 g ( x ) = − ( x − 4 ) 2 + 1

  3. h ( x ) = ( x + 1 ) 2 − 4 h ( x ) = ( x + 1 ) 2 − 4

  4. p ( x ) = − ( x + 1 ) 2 − 4 p ( x ) = − ( x + 1 ) 2 − 4

  5. q ( x ) = 2 ( x − 4 ) 2 + 1 q ( x ) = 2 ( x − 4 ) 2 + 1

  6. r ( x ) = ( x + 4 ) 2 − 1 r ( x ) = ( x + 4 ) 2 − 1

p ( x ) = − ( x + 1 ) 2 − 4 p ( x ) = − ( x + 1 ) 2 − 4

5.

Choose the equation that matches this graph.

...

  1. f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4

  2. g ( x ) = − ( x − 4 ) 2 + 1 g ( x ) = − ( x − 4 ) 2 + 1

  3. h ( x ) = ( x + 1 ) 2 − 4 h ( x ) = ( x + 1 ) 2 − 4

  4. p ( x ) = − ( x + 1 ) 2 − 4 p ( x ) = − ( x + 1 ) 2 − 4

  5. q ( x ) = 2 ( x − 4 ) 2 + 1 q ( x ) = 2 ( x − 4 ) 2 + 1

  6. r ( x ) = ( x + 4 ) 2 − 1 r ( x ) = ( x + 4 ) 2 − 1

q ( x ) = 2 ( x − 4 ) 2 + 1 q ( x ) = 2 ( x − 4 ) 2 + 1

6.

Choose the equation that matches this graph.

...

  1. f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4

  2. g ( x ) = − ( x − 4 ) 2 + 1 g ( x ) = − ( x − 4 ) 2 + 1

  3. h ( x ) = ( x + 1 ) 2 − 4 h ( x ) = ( x + 1 ) 2 − 4

  4. p ( x ) = − ( x + 1 ) 2 − 4 p ( x ) = − ( x + 1 ) 2 − 4

  5. q ( x ) = 2 ( x − 4 ) 2 + 1 q ( x ) = 2 ( x − 4 ) 2 + 1

  6. r ( x ) = ( x + 4 ) 2 − 1 r ( x ) = ( x + 4 ) 2 − 1

r ( x ) = ( x + 4 ) 2 − 1 r ( x ) = ( x + 4 ) 2 − 1

Video: Learning About Quadratic Equations and Graphs

Watch the following video to learn more about quadratic equations and graphs.

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Self Check

Which of the following equations matches the graph below?

  1. y = ( x + 3 ) 2 + 2
  2. y = − ( x + 3 ) 2 + 2
  3. y = − ( x − 3 ) 2 + 2
  4. y = ( x − 3 ) 2 + 2

Additional Resources

Matching Graphs and Equations

When matching graphs of quadratic functions and their equations, look for the matching vertex. Then look if the parabola is opening up or down. Finally, use another point to determine if the graph has been compressed or stretched. Recall this type of transformation will affect the a a -value of the equation.

Looking at the graph below, choose the equation that matches the graph.

Graph of a parabola on a coordinate plane. The x-axis has a scale of 1 extending from negative 10.5 to 10.5. The vertical-axis extends from approximately negative 10.5 to approximately 10.5 with a scale of 1.
  1. y = ( x − 2 ) 2 − 3 y = ( x − 2 ) 2 − 3
  2. y = ( x + 2 ) 2 − 3 y = ( x + 2 ) 2 − 3
  3. y = − ( x + 2 ) 2 − 3 y = − ( x + 2 ) 2 − 3
  4. y = − ( x − 2 ) 2 − 3 y = − ( x − 2 ) 2 − 3

Step 1 - Find the vertex.

The vertex is at ( − 2 , − 3 ) ( − 2 , − 3 ) .

Step 2 - Does the parabola open up or down?

The parabola opens up. So the leading coefficient is positive.

The correct answer is b. y = ( x + 2 ) 2 − 3 y = ( x + 2 ) 2 − 3 .

Remember the vertex form is y = a ( x − h ) 2 + k y = a ( x − h ) 2 + k . If the x x -variable is being added in the parenthesis, the parabola has been shifted left.

The ( x + 2 ) ( x + 2 ) moves the parabola left 2.

The -3 shifts it down 3 from the origin.

Step 3 - Use the vertex form of the equation and write the parts you know: the coordinates of the vertex. y = a ( x + 2 ) 2 − 3 y = a ( x + 2 ) 2 − 3

Step 4 - To solve for a a , find another point on the graph of the parabola and plug in those values for x x and y y in the equation you wrote.

Substitute the point ( − 1 , − 2 ) ( − 1 , − 2 ) into the equation to check the value of a a . We have hypothesized a = 1 a = 1 , and with the coordinates from the second point, x = − 1 x = − 1 and y = − 2 y = − 2 .

Step 5 - Solve for a a .

− 2 = a ( − 1 + 2 ) 2 − 3 − 2 = a ( − 1 + 2 ) 2 − 3 . Now solve for a a to find that a = 1 a = 1 .

Step 6 - Plug the value for a a back into the equation.

So, y = ( x + 2 ) 2 − 3 y = ( x + 2 ) 2 − 3 .

Try it

Matching Graphs and Equations

Match the graph with its equation in vertex form.

Graph of a parabola on a coordinate plane. The x-axis has a scale of 1 extending from negative 10.5 to 10.5. The vertical-axis extends from negative 10 to 10 with a scale of 1.

a. y = ( x + 1 ) 2 + 2 y = ( x + 1 ) 2 + 2

b. y = − ( x + 1 ) 2 + 2 y = − ( x + 1 ) 2 + 2

c. y = − ( x − 1 ) 2 + 2 y = − ( x − 1 ) 2 + 2

d. y = ( x − 1 ) 2 + 2 y = ( x − 1 ) 2 + 2

Here is how to match a graph and its equation:

Step 1 - Find the vertex. ( 1 , 2 ) ( 1 , 2 )

Step 2 - Does the parabola open up or down? Up — this means the leading coefficient is positive.

Step 3 - Use the vertex form of the equation and write the parts you know: the coordinates of the vertex. y = a ( x − 1 ) 2 + 2 y = a ( x − 1 ) 2 + 2

Step 4 - To solve for a a , find another point on the graph of the parabola and plug in those values for x x and y y in the equation you wrote. ( 0 , 3 ) ( 0 , 3 ) is also on the graph of this parabola. So, 3 = a ( 0 − 1 ) 2 + 2 3 = a ( 0 − 1 ) 2 + 2 .

Step 5 - Solve for a a .
3 = a ( 0 − 1 ) 2 + 2 3 = a ( 0 − 1 ) 2 + 2

3 = a ( − 1 ) 2 + 2 3 = a ( − 1 ) 2 + 2

3 = a + 2 3 = a + 2

1 = a 1 = a

Step 6 - Plug the value for
a a back into the equation. y = 1 ( x − 1 ) 2 + 2 y = 1 ( x − 1 ) 2 + 2
y = ( x − 1 ) 2 + 2 y = ( x − 1 ) 2 + 2

d.