Understanding the Behaviors of a Graph in Relation to Its Quadratic Expression

7.12.3 Understanding the Behaviors of a Graph in Relation to Its Quadratic Expression

7.12.3 • Understanding the Behaviors of a Graph in Relation to Its Quadratic Expression

Activity

1. Use the table with different values of $x$ to answer questions a–g for the values of $x^{2} + 10$ . (You may also use a spreadsheet tool, if available.)

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$x^{2} + 10$	a. ____	b. ____	c. ____	d. ____	e. ____	f. ____	g. ____

This table presents values for x and x^2 for integer inputs from -3 to 3, with a section to complete calculations for x^2 + 10.

a. When $x = - 3$ and $x^{2} = 9$ , what is the value of $x^{2} + 10$ ?

$(- 3)^{2} + 10 = 9 + 10 = 19$

b. When $x = - 2$ and $x^{2} = 4$ , what is the value of $x^{2} + 10$ ?

$(- 2)^{2} + 10 = 4 + 10 = 14$

c. When $x = - 1$ and $x^{2} = 1$ , what is the value of $x^{2} + 10$ ?

$(- 1)^{2} + 10 = 1 + 10 = 11$

d. When $x = 0$ and $x^{2} = 0$ , what is the value of $x^{2} + 10$ ?

$(0)^{2} + 10 = 0 + 10 = 10$

e. When $x = 1$ and $x^{2} = 1$ , what is the value of $x^{2} + 10$ ?

$(1)^{2} + 10 = 1 + 10 = 11$

f. When $x = 2$ and $x^{2} = 4$ , what is the value of $x^{2} + 10$ ?

$(2)^{2} + 10 = 4 + 10 = 14$

g. When $x = 3$ and $x^{2} = 9$ , what is the value of $x^{2} + 10$ ?

$(3)^{2} + 10 = 9 + 10 = 19$

2. Use the table with different values of $x$ to answer questions a–g for the values of $x^{2} - 3$ . (You may also use a spreadsheet tool, if available.)

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$x^{2} - 3$	a. ____	b. ____	c. ____	d. ____	e. ____	f. ____	g. ____

Table evaluating x^2 and x^2-3 for specified x values, with some results to be calculated.

a. When $x = - 3$ and $x^{2} = 9$ , what is the value of $x^{2} - 3$ ?

$(- 3)^{2} - 3 = 9 - 3 = 6$

b. When $x = - 2$ and $x^{2} = 4$ , what is the value of $x^{2} - 3$ ?

$(- 2)^{2} - 3 = 4 - 3 = 1$

c. When $x = - 1$ and $x^{2} = 1$ , what is the value of $x^{2} - 3$ ?

$(- 1)^{2} - 3 = 1 - 3 = - 2$

d. When $x = 0$ and $x^{2} = 0$ , what is the value of $x^{2} - 3$ ?

$(0)^{2} - 3 = 0 - 3 = - 3$

e. When $x = 1$ and $x^{2} = 1$ , what is the value of $x^{2} - 3$ ?

$(1)^{2} - 3 = 1 - 3 = - 2$

f. When $x = 2$ and $x^{2} = 4$ , what is the value of $x^{2} - 3$ ?

$(2)^{2} - 3 = 4 - 3 = 1$

g. When $x = 3$ and $x^{2} = 9$ , what is the value of $x^{2} + 10$ ?

$(3)^{2} - 3 = 9 - 3 = 6$

Select the solution button to see the solutions to questions 1 – 2 organized into a table.

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$x^{2} + 10$	19	14	11	10	11	14	19
$x^{2} - 3$	6	1	-2	-3	-2	1	6

Evaluation of algebraic expressions (x, x^2, x^2+10, x^2-3) for integer values of x ranging from -3 to 3.

3. In the earlier activity “Transformations with Quadratic Functions,” you observed the effects on the graph of adding or subtracting a constant term from $x^{2}$ . Study the values in the table created from questions 1 – 2 in this activity. Use these values to explain why the graphs in the earlier activity changed the way they did when a constant term is added or subtracted.

Compare your answer:

All values in the third row are 10 more than those in the second row, and all values in the last row are 3 less than those in the second row. This suggests that all the points that represent $x^{2} + 10$ are 10 units higher on the graph than those for $x^{2}$ , and all the points that represent $x^{2} - 3$ are 3 units lower than those for $x^{2}$ , which explains why the graphs move up or down when a constant term is added or subtracted.

4. Use the table with different values of $x$ to answer questions a–g for the values of $2 x^{2}$ . (You may also use a spreadsheet tool, if available.)

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$2 x^{2}$	a. _____	b. _____	c. _____	d. _____	e. _____	f. _____	g. _____

Evaluation of expressions x, x^2, and 2x^2 for integer values of x from -3 to 3, with the last row's values left for completion.

a. When $x = - 3$ and $x^{2} = 9$ , what is the value of $2 x^{2}$ ?

$2 (- 3)^{2} = 2 \cdot 9 = 18$

b. When $x = - 2$ and $x^{2} = 4$ , what is the value of $2 x^{2}$ ?

$2 (- 2)^{2} = 2 \cdot 4 = 8$

c. When $x = - 1$ and $x^{2} = 1$ , what is the value of $2 x^{2}$ ?

$2 (- 1)^{2} = 2 \cdot 1 = 2$

d. When $x = 0$ and $x^{2} = 0$ , what is the value of $2 x^{2}$ ?

$2 (0)^{2} = 2 \cdot 0 = 0$

e. When $x = 1$ and $x^{2} = 1$ , what is the value of $2 x^{2}$ ?

$2 (1)^{2} = 2 \cdot 1 = 2$

f. When $x = 2$ and $x^{2} = 4$ , what is the value of $2 x^{2}$ ?

$2 (2)^{2} = 2 \cdot 4 = 8$

g. When $x = 3$ and $x^{2} = 9$ , what is the value of $2 x^{2}$ ?

$2 (3)^{2} = 2 \cdot 9 = 18$

5. Use the table with different values of $x$ to answer questions a–g for the values of $\frac{1}{2} x^{2}$ . (You may also use a spreadsheet tool, if available.)

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$\frac{1}{2} x^{2}$	a. _____	b. _____	c. _____	d. _____	e. _____	f. _____	g. _____

Table of x, x^2, and (1/2)x^2 values for integers from -3 to 3, with (1/2)x^2 values to be determined.

a. When $x = - 3$ and $x^{2} = 9$ , what is the value of $\frac{1}{2} x^{2}$ ?

$\frac{1}{2} (- 3)^{2} = \frac{1}{2} \cdot 9 = \frac{9}{2} = 4 \frac{1}{2} = 4.5$

b. When $x = - 2$ and $x^{2} = 4$ , what is the value of $\frac{1}{2} x^{2}$ ?

$\frac{1}{2} (- 2)^{2} = \frac{1}{2} \cdot 4 = \frac{4}{2} = 2$

c. When $x = - 1$ and $x^{2} = 1$ , what is the value of $\frac{1}{2} x^{2}$ ?

$\frac{1}{2} (- 1)^{2} = \frac{1}{2} \cdot 1 = \frac{1}{2} = 0.5$

d. When $x = 0$ and $x^{2} = 0$ , what is the value of $\frac{1}{2} x^{2}$ ?

$\frac{1}{2} (0)^{2} = \frac{1}{2} \cdot 0 = 0$

e. When $x = 1$ and $x^{2} = 1$ , what is the value of $\frac{1}{2} x^{2}$ ?

$\frac{1}{2} (1)^{2} = \frac{1}{2} \cdot 1 = \frac{1}{2} = 0.5$

f. When $x = 2$ and $x^{2} = 4$ , what is the value of $\frac{1}{2} x^{2}$ ?

$\frac{1}{2} (2)^{2} = \frac{1}{2} \cdot 4 = \frac{4}{2} = 2$

g. When $x = 3$ and $x^{2} = 9$ , what is the value of $\frac{1}{2} x^{2}$ ?

$\frac{1}{2} (3)^{2} = \frac{1}{2} \cdot 9 = \frac{9}{2} = 4 \frac{1}{2} = 4.5$

6. Use the table with different values of $x$ to answer questions a–g for the values of $- 2 x^{2}$ . (You may also use a spreadsheet tool, if available.)

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$- 2 x^{2}$	a. _____	b. _____	c. _____	d. _____	e. _____	f. _____	g. _____

Values of x, x^2, and -2x^2 for integer x from -3 to 3, with the -2x^2 row values awaiting completion.

a. When $x = - 3$ and $x^{2} = 9$ , what is the value of $- 2 x^{2}$ ?

$- 2 (- 3)^{2} = - 2 \cdot 9 = - 18$

b. When $x = - 2$ and $x^{2} = 4$ , what is the value of $- 2 x^{2}$ ?

$- 2 (- 2)^{2} = - 2 \cdot 4 = - 8$

c. When $x = - 1$ and $x^{2} = 1$ , what is the value of $- 2 x^{2}$ ?

$- 2 (- 1)^{2} = - 2 \cdot 1 = - 2$

d. When $x = 0$ and $x^{2} = 0$ , what is the value of $- 2 x^{2}$ ?

$- 2 (0)^{2} = - 2 \cdot 0 = 0$

e. When $x = 1$ and $x^{2} = 1$ , what is the value of $- 2 x^{2}$ ?

$- 2 (1)^{2} = - 2 \cdot 1 = - 2$

f. When $x = 2$ and $x^{2} = 4$ , what is the value of $- 2 x^{2}$ ?

$- 2 (2)^{2} = - 2 \cdot 4 = - 8$

g. When $x = 3$ and $x^{2} = 9$ , what is the value of $- 2 x^{2}$ ?

$- 2 (3)^{2} = - 2 \cdot 9 = - 18$

Select the solution button to see the solutions to questions 4 – 6 organized into a table.

$x$	-3	-2	-1	1	2	3
$x^{2}$	9	4	1	1	4	9
$2 x^{2}$	18	8	2	2	8	18
$\frac{1}{2} x^{2}$	$4 \frac{1}{2} = 4.5$	2	$\frac{1}{2} = 0.5$	$\frac{1}{2} = 0.5$	2	$4 \frac{1}{2} = 4.5$
$- 2 x^{2}$	-18	-8	-2	-2	-8	-18

This table displays the results of evaluating various quadratic expressions, including x, x^2, 2x^2, (1/2)x^2, and -2x^2, for integer x-values ranging from -3 to 3.

7. In the earlier activity “Transformations with Quadratic Functions,” you observed the effects on the graph of multiplying $x^{2}$ by different coefficients. Study the values in the table created from questions 1 and 2 in this activity. Use these values to explain why the graphs in the earlier activity changed the way they did when $x^{2}$ is multiplied by a number greater than 1, by a negative number less than or equal to -1, and by numbers between -1 and 1.

Compare your answer:

When we multiply $x^{2}$ by the positive coefficient 2, all the output values double. For any input value $x$ , the output value of $2 x^{2}$ is twice as large as that of $x^{2}$ . So, the width of the graph becomes narrower because it grows faster. When we multiply $x^{2}$ by a number between 0 and 1, the output values become smaller, so the graph becomes wider or vertically compressed. When we multiply by a negative number, all the positive values become negative, which reflects the direction of the opening of the graph from opening up to opening down.

Video: Understanding the Behaviors of a Graph in Relation to Its Quadratic Expression

Watch the following video to learn more about the behaviors of a graph in relation to its quadratic expression.

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Self Check

Which curve is narrower than

f(x)=x^2

$f(x)=(x+3)^2$
$f(x)=\frac{1}{3}x^2$
$f(x)=3x^2$
$f(x)=x^2+3$

Additional Resources

Comparing Transformations to Quadratic Graphs

Remember that the graph representing any quadratic function is a shape called a parabola. People often say that a parabola "opens upward" when the lowest point on the graph is the vertex (where the graph changes direction), and "opens downward" when the highest point on the graph is the vertex. Each coefficient in a quadratic expression written in standard form $a x^{2} + b x + c$ tells us something important about the graph that represents it.

The graph of $f (x) = x^{2}$ is a parabola opening upward with vertex at $(0, 0)$ . Adding a constant term 5 gives $f (x) = x^{2} + 5$ and raises the graph by 5 units. Subtracting 4 from $x^{2}$ gives $f (x) = x^{2} - 4$ and moves the graph 4 units down.

Graph of the parent quadratic function is given in black on the coordinate plane. Functions that have been vertically translated are graphed and labeled as y equals x squared plus 5 and y equals x squared minus four.

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$x^{2} + 5$	14	9	6	5	6	9	14
$x^{2} - 4$	5	0	-3	-4	-3	0	5

Values of algebraic expressions (x, x^2, x^2+5, x^2-4) for integer values of x from -3 to 3.

A table of values can help us see that adding 5 to $x^{2}$ increases all the output values of $f (x) = x^{2}$ by 5, which explains why the graph moves up 5 units. Subtracting 4 from $x^{2}$ decreases all the output values of $f (x) = x^{2}$ by 4, which explains why the graph shifts down by 4 units.

In general, the constant term of a quadratic expression in standard form influences the vertical position of the graph. An expression with no constant term (such as $x^{2}$ or $x^{2} + 9 x$ ) means that the constant term is 0, so the $y$ -intercept of the graph is on the $x$ -axis. It's not shifted up or down relative to the $x$ -axis.

Coefficient of the Squared Term

The coefficient of the squared term in a quadratic function also tells us something about its graph. The coefficient of the squared term in $f (x) = x^{2}$ is 1. Its graph is a parabola that opens upward.

Multiplying $x^{2}$ by a number greater than 1 makes the graph steeper, so the parabola is narrower than that representing $x^{2}$ .

Multiplying $x^{2}$ by a number less than 1 but greater than 0 makes the graph less steep, so the parabola is wider than that representing $x^{2}$ .

Multiplying $x^{2}$ by a number less than 0 makes the parabola open downward.

Graph of the parent quadratic function is given in black on the coordinate plane. Functions that have been vertically dilated are graphed and labeled as y equals 2 times x squared and y equals one-half times x squared. The parabola that opens down is labeled y equals negative 2 times x squared.

$x$	-3	-2	-1	1	2	3
$x^{2}$	9	4	1	1	4	9
$2 x^{2}$	18	8	2	2	8	18
$- 2 x^{2}$	-18	-8	-2	-2	-8	-18

Evaluations of x, x^2, 2x^2, and -2x^2 for integer x values ranging from -3 to 3.

If we compare the output values of $2 x^{2}$ and $- 2 x^{2}$ , we see that they are opposites, which suggests that one graph would be a reflection of the other across the $x$ -axis.

In the previous examples, parameters were being applied to the output values of the function. In other words, after an input value was squared, something was then added, subtracted, or multiplied with the squared value. The resulting impact on the graph was to vertically alter it.

However, what happens when the input value is altered before applying the exponent? In these situations, the impact on the graph is to produce a horizontal transformation.

Horizontal Shifts

Return to the quadratic parent function, $f (x) = x^{2}$ . The function is equivalent to $f (x) = (x - 0)^{2}$ . While a bit obvious, this implies there is no change to the input value before being squared.

If the number used to change the input is greater than 0, then the graph of the function is shifted to the right.

Conversely, if the number used to change the input is less than 0, the graph of the function is shifted to the left. Note the impact to the subtraction operation when negative values are subtracted from the input.

Graph of the parent quadratic function is given in black on the coordinate plane. Functions that have been horizontally translated are graphed and labeled as y equals the quantity of x plus three squared and y equals the quantity of x minus three squared.

$x$	-3	-2	-1	0	1	2	3
$x^{2}$	9	4	1	0	1	4	9
$(x - 3)^{2}$	36	25	16	9	4	1	0
( $x - - 3)^{2} = (x + 3)^{2}$	0	1	4	9	16	25	36

Evaluations of algebraic expressions, including x, x^2, (x-3)^2, and (x+3)^2, for integer values of x ranging from -3 to 3.

Notice in this table of values that the change to the input values takes place before squaring the term. The constant is directly affecting the $x$ -value before applying the function rule.

Examine the vertex of each graph. When we compare the graphs of $(x - 3)^{2}$ and $(x + 3)^{2}$ to the graph of the parent function, we see the parabola has been shifted to the right 3 units and left 3 units, respectively. Thus, a horizontal shift in the graph results when a constant has been subtracted from the input value.

Horizontal Dilations and Reflections

Now we will investigate the impact of a coefficient on the input value.

Multiplying the input value by a value greater than 1 and then applying the exponent, results in a parabola that has been horizontally compressed.

Multiplying the input value by a value that is between 0 and 1, yields a parabola that has been horizontally stretched.

Graph of the parent quadratic function is given in black on the coordinate plane. Functions that have been horizontally dilated are graphed and labeled as y equals one-half times x quantity squared and y equals two times x quantity squared.

$x$	-3	-2	-1	1	2	3
$x^{2}$	9	4	1	1	4	9
$(2 x)^{2}$	36	16	4	4	16	36
$(\frac{1}{2} x)^{2}$	2.25	1	0.25	0.25	1	2.25
$(- 2 x)^{2}$	36	16	4	4	16	36

Evaluation of quadratic functions x^2, (2x)^2, (1/2x)^2, and (-2x)^2 for integer x values from -3 to 3.

If the input value is multiplied by a negative number, then the parabola is reflected over the $y$ -axis. However, due to the symmetric nature of parabolas, this is difficult to discern visually.

Graph of the parent quadratic function is given in black on the coordinate plane. The function y equals negative 2 times x quantity squared is also graphed and labeled.

Multiplying the input value by a number changes the quantity that is squared. As a result, the graph of the equation is horizontally stretched or compressed.

Try it

Comparing Transformations to Quadratic Graphs

If the red graph (in the middle) is the parent function, $f (x) = x^{2}$ , what are the equations of the blue graph (at the top) and green graph (at the bottom)?

Graph of the parent quadratic function is given in red on the coordinate plane. Two other parabolas are also graphed. Both the x-axis and y-axis extend from negative 10 to 10 with a scale of 1.

Here is how to solve to find the equations of the other graphs: The green graph is the same as $f (x) = x^{2}$ but is shifted down 2. The function will be $f (x) = x^{2} - 2$ . The blue graph is the same as $f (x) = x^{2}$ but is shifted up 4. The function will be $f (x) = x^{2} + 4$ .

If the red graph (in the middle) is the parent function, $f (x) = x^{2}$ , what are the equations of the blue graph (to the right) and green graph (to the left)?

Graph of the parent quadratic function is given in red on the coordinate plane. Two other parabolas are also graphed. The x-axis extends from negative 6 to 6 with a scale of 0.5 while the y-axis extends from negative 3 to 7 with a scale of 0.5.

Here is how to solve to find the equations of the other graphs.

The green graph is the same as $f (x) = x^{2}$ but is shifted left $\frac{3}{2}$ . The function will be $f (x) = (x + \frac{3}{2})^{2}$ .

The blue graph is the same as $f (x) = x^{2}$ but is shifted right 3. The function will be $f (x) = (x - 3)^{2}$ .

The red graph (in the middle) is $f (x) = x^{2}$ . The equations of the other graphs are $f (x) = \frac{1}{3} x^{2}$ and $f (x) = 5 x^{2}$ .

Which function matches the green graph (on the inside)? The blue graph (on the outside)?

The green graph is narrower than $f (x) = x^{2}$ . It must have a coefficient that is larger than the coefficient of $x^{2}$ . The green graph is $f (x) = 5 x^{2}$ .

The blue graph is wider than $f (x) = x^{2}$ . It must have a coefficient that is smaller than the coefficient of $x^{2}$ . The blue graph is $f (x) = \frac{1}{3} x^{2}$ .

The black graph (in the middle) is the parent function $f (x) = x^{2}$ . The equations of the other graphs are $f (x) = (\frac{2}{5} x)^{2}$ and $f (x) = (\frac{4}{3} x)^{2}$ . Which function matches the green graph (on the inside)? The blue graph (on the outside)?

Graph of the parent quadratic function is given in black on the coordinate plane. Two other parabolas are also graphed. The x-axis extends from negative 5 to 5 with a scale of 1 while the y-axis extends from negative 4 to 9 with a scale of 1. The green parabola lies inside of the parent function while the blue parabola lies outside of the path of the parent function.

Compare your answer:

Here is how to solve to find the equations of the other graphs:

The green graph has been horizontally compressed in comparison to the parent function. Therefore, the coefficient impacting the input value must be greater than 1. That means the green graph represents $f (x) = (\frac{4}{3} x)^{2}$ .

The blue graph has been horizontally stretched in comparison to the parent function. So, the coefficient impacting the input value must be greater than 0 but less than 1. That means the blue graph represents $f (x) = (\frac{2}{5} x)^{2}$ .