Almost all transformed functions can be written like this:

$\displaystyle g(x)=a[f(b(x-c))]+d$

where $\displaystyle f(x)$ is the parent function. In this case, our parent function is $\displaystyle f(x)=x^{2}$, so we can write $\displaystyle g(x)$ this way:

$\displaystyle g(x)=a[b(x-c)]^{2}+d$

Luckily, for this problem, we only have to worry about $\displaystyle a$ and $\displaystyle d$.

$\displaystyle a$ represents the vertical stretch factor of the graph.

• If $\displaystyle |a|$ is less than 1, the graph has been vertically compressed by a factor of $\displaystyle |a|$. It's almost as if someone squished the graph while their hands were on the top and bottom. This would make a parabola, for example, look wider.
• If $\displaystyle |a|$ is greater than 1, the graph has been vertically stretched by a factor of $\displaystyle |a|$. It's almost as if someone pulled on the graph while their hands were grasping the top and bottom. This would make a parabola, for example, look narrower.

$\displaystyle d$ represents the vertical translation of the graph.

• If $\displaystyle d$ is positive, the graph has been shifted up $\displaystyle d$ units.
• If $\displaystyle d$ is negative, the graph has been shifted down $\displaystyle d$ units.

For this problem, $\displaystyle a$ is 4 and $\displaystyle d$ is -3, causing vertical stretch by a factor of 4 and a vertical translation down 3 units.

Begin with the standard equation for a parabola: $\displaystyle f(x)=x^2$.

Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the $\displaystyle x$ term. To shift 4 units to the right, subtract 4 within the parenthesis.

$\displaystyle f(x)=(x-4)^2$

The width of the parabola is determined by the magnitude of the coefficient in front of $\displaystyle x$. To make a parabola narrower, use a whole number coefficient. Halving the width indicates a coefficient of two.

$\displaystyle f(x) = 2(x-4)^2$

Begin with the standard equation for a parabola: $\displaystyle f(x)=x^2$.

Vertical shifts to this standard equation are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift upward 2 units, add 2.

$\displaystyle f(x)=x^2+2$

Begin with the standard equation for a parabola: $\displaystyle f(x)=x^2$.

Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the $\displaystyle x$. If the coefficient is negative, then the parabola opens downward.

$\displaystyle f(x)=-x^2$

The width of the parabola is determined by the magnitude of the coefficient in front of $\displaystyle x$. To make a parabola wider, use a fractional coefficient. Doubling the width indicates a coefficient of one-half.

$\displaystyle f(x)=-(\frac{1}{2})x^2=-0.5x^2$

Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift upward 3 units, add 3.

$\displaystyle f(x)=-0.5x^2 +3$

Begin with the standard equation for a parabola: $\displaystyle f(x)=x^2$.

Vertical shifts to this standard equation are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 5 units, subtract 5.

$\displaystyle f(x)=x^2-5$

Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the $\displaystyle x$ term. To shift 6 units to the right, subtract 6 within the parenthesis.

$\displaystyle f(x)=(x-6)^2 - 5$

Begin with the standard equation for a parabola: $\displaystyle f(x)=x^2$.

Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the $\displaystyle x$ term. To shift 12 units to the right, subtract 12 within the parenthesis.

$\displaystyle f(x) = (x-12)^2$

Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the $\displaystyle x$. If the coefficient is negative, then the parabola opens downward.

$\displaystyle f(x) = -(x-12)^2$

Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 4 units, subtract 4.

$\displaystyle f(x) = -(x-12)^2 - 4$

Begin with the standard equation for a parabola: $\displaystyle f(x)=x^2$.

Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the $\displaystyle x$ term. To shift 4 units to the left, add 4 within the parenthesis.

$\displaystyle f(x) = (x+4)^2$

The width of the parabola is determined by the magnitude of the coefficient in front of $\displaystyle x$. To make a parabola wider, use a fractional coefficient. Doubling the width indicates a coefficient of one-fourth.$\displaystyle f(x) = \frac{1}{4}(x+4)^2=0.25(x+4)^2$

Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 5 units, subtract 5.

$\displaystyle f(x) = 0.25(x+4)^2 - 5$

Shifting $\displaystyle y=1-x$ left 2 units will change the y-intercept from $\displaystyle (0,1)$ to $\displaystyle (0.-1)$.

The new equation after shifting left 2 units is:

$\displaystyle y=1-(x+2)=-1-x$

Shifting up 3 units will add 3 to the y-intercept of the new equation.

The answer is:  $\displaystyle y=2-x$

It helps to evaluate the expression algebraically.

$\displaystyle -f(x)=-x^2$. Any time you multiply a function by a -1, you reflect it over the x axis. It helps to graph for verification.

This is the graph of $\displaystyle x^2$

and this is the graph of $\displaystyle -x^2$

Algebraically, $\displaystyle f(-x)=e^{-x}$.

This is a reflection across the y axis.

This is the graph of $\displaystyle e^x$

And this is the graph of $\displaystyle e^{-x}$