Functions and Graphs - Math

To solve for the -intercept, set to zero and solve for :

To solve for the -intercept, set to zero and solve for :

Slope-intercept form is .

Since the equation given in the question is based off of the parent function , we can write the general form for transformations like this:

determines the vertical stretch or compression factor.

• If is greater than 1, the function has been vertically stretched (expanded) by a factor of .
• If is between 0 and 1, the function has been vertically compressed by a factor of .

In this case, is 4, so the function has been vertically stretched by a factor of 4.

determines the horizontal stretch or compression factor.

• If is greater than 1, the function has been horizontally compressed by a factor of .
• If is between 0 and 1, the function has been horizontally stretched (expanded) by a factor of .

In this case, is 6, so the function has been horizontally compressed by a factor of 6. (Remember that horizontal stretch and compression are opposite of vertical stretch and compression!)

determines the horizontal translation.

• If is positive, the function was translated units right.
• If is negative, the function was translated units left.

In this case, is 3, so the function was translated 3 units right.

determines the vertical translation.

• If is positive, the function was translated units up.
• If is negative, the function was translated units down.

In this case, is -7, so the function was translated 7 units down.

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain .

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain

Use substution to solve this problem:

becomes and then is substituted into the second equation. Then solve for :

, so and to give the solution .

To solve for the -intercept, set to zero and solve for :

To solve for the -intercept, set to zero and solve for :

Slope-intercept form is .

Since the equation given in the question is based off of the parent function , we can write the general form for transformations like this:

determines the vertical stretch or compression factor.

• If is greater than 1, the function has been vertically stretched (expanded) by a factor of .
• If is between 0 and 1, the function has been vertically compressed by a factor of .

In this case, is 4, so the function has been vertically stretched by a factor of 4.

determines the horizontal stretch or compression factor.

• If is greater than 1, the function has been horizontally compressed by a factor of .
• If is between 0 and 1, the function has been horizontally stretched (expanded) by a factor of .

In this case, is 6, so the function has been horizontally compressed by a factor of 6. (Remember that horizontal stretch and compression are opposite of vertical stretch and compression!)

determines the horizontal translation.

• If is positive, the function was translated units right.
• If is negative, the function was translated units left.

In this case, is 3, so the function was translated 3 units right.

determines the vertical translation.

• If is positive, the function was translated units up.
• If is negative, the function was translated units down.

In this case, is -7, so the function was translated 7 units down.

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain .

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain

Use substution to solve this problem:

becomes and then is substituted into the second equation. Then solve for :

, so and to give the solution .

To solve for the -intercept, set to zero and solve for :

To solve for the -intercept, set to zero and solve for :

Slope-intercept form is .

Since the equation given in the question is based off of the parent function , we can write the general form for transformations like this:

determines the vertical stretch or compression factor.

• If is greater than 1, the function has been vertically stretched (expanded) by a factor of .
• If is between 0 and 1, the function has been vertically compressed by a factor of .

In this case, is 4, so the function has been vertically stretched by a factor of 4.

determines the horizontal stretch or compression factor.

• If is greater than 1, the function has been horizontally compressed by a factor of .
• If is between 0 and 1, the function has been horizontally stretched (expanded) by a factor of .

In this case, is 6, so the function has been horizontally compressed by a factor of 6. (Remember that horizontal stretch and compression are opposite of vertical stretch and compression!)

determines the horizontal translation.

• If is positive, the function was translated units right.
• If is negative, the function was translated units left.

In this case, is 3, so the function was translated 3 units right.

determines the vertical translation.

• If is positive, the function was translated units up.
• If is negative, the function was translated units down.

In this case, is -7, so the function was translated 7 units down.

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain .

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain

Use substution to solve this problem:

becomes and then is substituted into the second equation. Then solve for :

, so and to give the solution .

To solve for the -intercept, set to zero and solve for :

To solve for the -intercept, set to zero and solve for :