11.5 Transformations of Functions

Describe the single transformation that maps $y = x^2$ to $y = (x + 5)^2$.

Describe the single transformation that maps $y = f(x)$ to $y = f(x) + 4$.

Describe the transformation mapping $y = f(x)$ to $y = -f(x)$.

Describe the transformation mapping $y = f(x)$ to $y = 3 \cdot f(x)$.

The graph of $y = f(x)$ has $y$-intercept $(0, 4)$. State the $y$-intercept of $y = f(x) - 3$.

The graph of $y = f(x)$ has minimum at $(2, -3)$. State the new minimum of $y = f(x - 4)$.

The graph of $y = f(x)$ contains the point $(2, 5)$. What point does the graph of $y = f(-x)$ contain?

State the vertex of $y = (x - 2)^2 + 5$.

Describe the transformation mapping $y = f(x)$ to $y = f(2x)$.

Describe the single transformation from $y = \sqrt{x}$ to $y = \sqrt{x} + 3$.

The graph of $y = x^2$ is translated 3 units right and 2 units up. Write the equation of the new graph.

The graph of $y = x^2$ is reflected in the $x$-axis and translated 4 units up. Write the new equation.

Describe the single transformation mapping $y = x^2$ to $y = 3x^2$.

The graph of $y = f(x)$ has minimum at $(-1, 4)$. State the coordinates of the new minimum for $y = f(x) - 5$.

The graph of $y = f(x)$ has minimum at $(-1, 4)$ and $y$-intercept $(0, 7)$. State the new turning point and $y$-intercept of $y = -f(x)$, and say whether it becomes a maximum or minimum.

The graph of $y = f(x)$ contains the points $(0, 2)$ and $(4, -1)$. State the points after the transformation $y = 2f(x)$.

Starting from $y = x^2$, translate 2 left, then reflect in the $x$-axis. Write the final equation.

The graph of $y = f(x)$ contains $(6, 8)$. State the point on the graph of $y = f(2x)$ corresponding to this.

The graph of $y = f(x)$ has a maximum at $(2, 5)$. State the maximum of $y = f(x - 1) + 3$.

The graph of $y = \frac{1}{x}$ has a vertical asymptote at $x = 0$. State the vertical asymptote of $y = \frac{1}{x - 3}$.

Starting from $y = x^2$, reflect in the $x$-axis, then translate 2 right and 4 up. State the final equation and the vertex.

Describe a sequence of transformations from $y = x^2$ to $y = 2(x - 3)^2 - 1$.

If $f(x) = x^2$, and $g(x) = f(x - 1) + 4$, find the $x$-values for which $g(x) = 8$.

The point $(2, 3)$ is on the graph of $y = f(x)$. State the corresponding point on the graph of $y = -f(x) + 4$.

The graph of $y = f(x)$ has key features at $x = 1$ and $y$-value 3. State the corresponding features on $y = 2f(3x)$.

A parabola has vertex $(2, -1)$ and passes through $(0, 7)$. Find its equation in vertex form.

The graph of $y = f(x)$ has $x$-intercepts at $x = 1$ and $x = 5$, and $y$-intercept $(0, -5)$. State the corresponding intercepts of $y = -f(x)$.

The graph of $y = g(x)$ is obtained from $y = f(x)$ by a reflection in the $y$-axis followed by a translation 2 units down. Write $g(x)$ in terms of $f$.

Express $y = 2x^2 + 12x + 13$ in vertex form, then describe the chain of transformations from $y = x^2$.

The graph of $y = f(x)$ has its maximum at $(0, 4)$. Where is the maximum of $y = f(2(x - 3))$?

The graph of $y = f(x)$ has minimum at $(2, -3)$. Find the new minimum after: shift right 3, then reflect in the $x$-axis, then shift up 5.

The graph of $y = g(x)$ is obtained from $y = f(x)$ by translating 3 right then stretching vertically by factor 2. Write the inverse transformation chain (from $g$ back to $f$).

A bounded region between $y = f(x)$ and the $x$-axis has area 12. State the area of the region between $y = 3 f(x/2)$ and the $x$-axis.

A function $g(x) = (x + 3)^2 - 4$ is obtained from $y = x^2$ by a sequence of transformations. State the sequence (in order).

The graph of $y = \sin x$ is transformed so that its amplitude becomes 3, its period is $\pi$, and it is shifted up by 2. Write the equation.

Let $f(x) = (x - 2)^2$ and $g(x) = f(x + 4) - 1$. Find the vertex of $y = g(x)$ and write $g$ in expanded form.

The graph of $y = f(x)$ passes through $(1, 2)$ and $(3, 8)$. State the corresponding points on $y = f(x - 2) + 5$, and find the average rate of change of the new function between them.

The function $y = x^2$ is translated so that its new vertex is $(4, -7)$. Find $h$ and $k$ if the new equation is $y = (x - h)^2 + k$.

The graph of $y = f(x)$ passes through $(2, 5)$. State the corresponding point on the graph of $y = f^{-1}(x)$, and describe the geometric relationship.

Is the function $f(x) = x^3 - 4x$ even, odd, or neither? After a translation 2 units right, is the resulting function even, odd, or neither?