Question

Given the function \(y=f(x),\) write the equation of the form \(y-k=a f(b(x-h))\) that would result from each combination of transformations. a) a vertical stretch about the \(x\) -axis by a factor of \(3,\) a reflection in the \(x\) -axis, a horizontal translation of 4 units to the left, and a vertical translation of 5 units down b) a horizontal stretch about the \(y\) -axis by a factor of \(\frac{1}{3},\) a vertical stretch about the \(x\) -axis by a factor of \(\frac{3}{4},\) a reflection in both the \(x\) -axis and the \(y\) -axis, and a translation of 6 units to the right and 2 units up

Short answer

a) y = -3f(x + 4) - 5 b) y = -3/4f(-3(x - 6)) + 2

Step-by-step solution

Understand the transformations

Transformations include vertical stretches/compressions, horizontal stretches/compressions, reflections, and translations. Vertical changes affect the 'a' and 'k' values in the equation, while horizontal changes affect the 'b' and 'h' values.

Identify transformations for part (a)

For part (a), we have a vertical stretch by a factor of 3, a reflection in the x-axis, a horizontal translation 4 units left, and a vertical translation 5 units down.

Apply vertical stretch and reflection (part a)

The vertical stretch by a factor of 3 is represented by 'a = 3'. The reflection in the x-axis changes the sign of 'a': a = -3y = -3f(x).

Apply horizontal translation (part a)

A horizontal translation 4 units to the left is represented by 'h = -4': y = -3f(x + 4).

Apply vertical translation (part a)

A vertical translation 5 units down is represented by 'k = -5': y + 5 = -3f(x + 4).

Write the final equation for part (a)

The final equation is: y + 5 = -3f(x + 4) Which simplifies to: y = -3f(x + 4) - 5.

Identify transformations for part (b)

For part (b), we have a horizontal stretch by a factor of 1/3, a vertical stretch by a factor of 3/4, reflections in both the x-axis and the y-axis, a horizontal translation 6 units right, and a vertical translation 2 units up.

Apply horizontal stretch and reflection (part b)

The horizontal stretch by a factor of 1/3 is represented by 'b = 3'. The reflection in the y-axis changes the sign of 'b': b = -3y = f(-3x).

Apply vertical stretch and reflection (part b)

The vertical stretch by a factor of 3/4 is represented by 'a = 3/4'. The reflection in the x-axis changes the sign of 'a': a = -3/4y = -3/4f(-3x).

Apply horizontal translation (part b)

A horizontal translation of 6 units to the right is represented by 'h = 6': y = -3/4f(-3(x - 6)).

Apply vertical translation (part b)

A vertical translation of 2 units up is represented by 'k = 2': y - 2 = -3/4f(-3(x - 6)).

Write the final equation for part (b)

The final equation is: y - 2 = -3/4f(-3(x - 6)) Which simplifies to: y = -3/4f(-3(x - 6)) + 2.

Key concepts

Vertical Stretch

A vertical stretch occurs when we multiply the entire function by a constant factor, making the graph taller. Imagine pulling the graph upward from its lowest point. If the constant factor is greater than 1, the function gets taller; if it's between 0 and 1, it gets shorter.

For example, if we have a function defined as \( y = f(x) \) and we want to apply a vertical stretch by a factor of 3, the transformed function becomes \( y = 3f(x) \). This changes the amplitude of the graph, stretching it away from the x-axis.

Specifically in exercise part (a), applying a vertical stretch by a factor of 3 makes our function \( y = 3f(x) \). If we have more combinations of transformations, ensure to appropriately combine them while maintaining the vertical stretch factor.

Horizontal Translation

Horizontal translation shifts the graph of the function left or right. It doesn’t change the shape of the graph, only its position along the x-axis.

Suppose we have the function \( y = f(x) \) and we translate it horizontally by 4 units to the left. The transformed function becomes \( y = f(x + 4) \). Notice the 'plus' sign: moving left effectively means adding inside the function's argument—but it makes the graph shift to the left.

In exercise part (a), we had a horizontal translation of 4 units to the left represented by changing the argument of the function accordingly in our equation. By the same principle, in part (b) we applied a horizontal shift 6 units to the right by modifying the argument like so \( y = f(x - 6) \). This helps in positioning the graph in its desired location on the x-axis.

Reflection

Reflection flips the graph of the function over a specific axis, either the x-axis or the y-axis.

Reflecting over the x-axis multiplies the function by -1. For a function \( y = f(x) \), flipping it over the x-axis gives \( y = -f(x) \). This inverts all y-values, turning highs into lows and vice versa.

Reflecting over the y-axis changes the input of the function. For a function \( y = f(x) \), reflecting it over the y-axis means \( y = f(-x) \). Each positive x becomes negative, and each negative x becomes positive.

In exercise part (a), reflecting the function over the x-axis resulted in changing the sign of the entire function: \( y = -3f(x + 4) \). Meanwhile, in part (b), we reflected over both the x-axis and the y-axis. This meant incorporating both transformations in our equations.

Vertical Translation

Vertical translation shifts the graph of the function up or down. Similar to horizontal translation, it alters the graph's position without affecting its shape.

For a function \( y = f(x) \), moving it 5 units down results in \( y = f(x) - 5 \). We subtract 5 from the entire function because we are shifting vertically downward.

Respectively, moving a function up adds to the function. For example, if the function moves 2 units up, it transforms to \( y = f(x) + 2 \).

In the given exercise, part (a) involved vertically translating the graph 5 units down, which added -5 to the entire function, resulting in \( y = -3f(x + 4) - 5 \). Part (b) involved moving the graph 2 units up, adding 2 to the entire function, giving \( y = -\frac{3}{4} f(-3(x - 6)) + 2 \).