Question

For each transformation, identify the values of \(h\) and \(k .\) Then, write the equation of the transformed function in the form \(y-k=f(x-h)\) . a) \(f(x)=\frac{1}{x},\) translated 5 units to the left and 4 units up b) \(f(x)=x^{2},\) translated 8 units to the right and 6 units up c) \(f(x)=|x|,\) translated 10 units to the right and 8 units down d) \(y=f(x),\) translated 7 units to the left and 12 units down

Short answer

(a) \( y - 4 = \frac{1}{x + 5} \), (b) \( y - 6 = (x - 8)^2 \), (c) \( y + 8 = |x - 10| \), (d) \( y + 12 = f(x + 7) \)

Step-by-step solution

Identify values of h and k for part a

The function given is \(f(x)=\frac{1}{x}\). A translation of 5 units to the left means \( h = -5 \). A translation of 4 units up means \( k = 4 \).

Write the equation for part a

Using the values of \( h \) and \( k \), the transformed function is \( y - 4 = \frac{1}{x + 5} \).

Identify values of h and k for part b

The function given is \(f(x)=x^2\). A translation of 8 units to the right means \( h = 8 \). A translation of 6 units up means \( k = 6 \).

Write the equation for part b

Using the values of \( h \) and \( k \), the transformed function is \( y - 6 = (x - 8)^2 \).

Identify values of h and k for part c

The function given is \(f(x)=|x|\). A translation of 10 units to the right means \( h = 10 \). A translation of 8 units down means \( k = -8 \).

Write the equation for part c

Using the values of \( h \) and \( k \), the transformed function is \( y + 8 = |x - 10| \).

Identify values of h and k for part d

The function given is \(y=f(x)\). A translation of 7 units to the left means \( h = -7 \). A translation of 12 units down means \( k = -12 \).

Write the equation for part d

Using the values of \( h \) and \( k \), the transformed function is \( y + 12 = f(x + 7) \).

Key concepts

Translations in the Coordinate Plane

Translations in the coordinate plane involve shifting a function's graph horizontally, vertically, or both. When we translate a function, we simultaneously slide it along the x-axis (horizontal shift) and/or the y-axis (vertical shift) without changing its shape.

For example, translating a function 5 units left and 4 units up means every point on the graph moves horizontally left by 5 units and vertically up by 4 units. This is a common task in function transformations, and once you understand how to handle these translations, solving these problems becomes straightforward.

Transformations of Functions

Transformations of functions include translations, reflections, stretches, and compressions. Translations are the most common types that involve shifting the graph of a function horizontally or vertically. Understanding transformations allows us to take a basic function like a parabola, an absolute value function, or a reciprocal function and modify its position on the graph.

To translate a function, we adjust its equation by changing the variables inside and outside the function. For example, if we have a function, say, \( f(x) \), translating it results in a new function \( y - k = f(x - h) \), where \( h \) and \( k \) are the horizontal and vertical shifts respectively. The signs of \( h \) and \( k \) indicate the direction of the shift.

Horizontal and Vertical Shifts

Horizontal and vertical shifts are specific types of translations that affect where the graph of a function appears on the coordinate plane. A horizontal shift moves the function left or right along the x-axis, while a vertical shift moves it up or down the y-axis.

Horizontal shifts are determined by the value of \( h \):

• If \( h > 0 \), the graph shifts to the right.
• If \( h < 0 \), the graph shifts to the left.

Similarly, vertical shifts depend on \( k \):

• If \( k > 0 \), the graph shifts upwards.
• If \( k < 0 \), the graph shifts downwards.

For example, translating \( f(x) = x^2 \) eight units right and six units up involves altering the equation to \( y - 6 = (x - 8)^2 \). Each translation can be analyzed separately and involves straightforward adjustments to the function's formula.