Question

The key point (-12,18) is on the graph of \(y=f(x) .\) What is its image point under each transformation of the graph of \(f(x) ?\) a) \(y+6=f(x-4)\) b) \(y=4 f(3 x)\) c) \(y=-2 f(x-6)+4\) d) \(y=-2 f\left(-\frac{2}{3} x-6\right)+4\) e) \(y+3=-\frac{1}{3} f(2(x+6))\)

Short answer

a) (-8, 12), b) (-4, 72), c) (-6, -32), d) (9, -32), e) (-3, -9)

Step-by-step solution

Identifying the Point

The original point given is (-12, 18). We will use this as the basis to find the transformed points for the different equations.

Transformation for (a)

The transformation is given by: \(y+6=f(x-4)\). The transformation involves shifting the graph horizontally by 4 units to the right and vertically by 6 units down.- New x-coordinate: \(x' = x + 4 = -12 + 4 = -8\)- New y-coordinate: \(y' = y - 6 = 18 - 6 = 12\)Thus, the image point is (-8, 12).

Transformation for (b)

The transformation is given by: \(y=4 f(3 x)\). This involves a horizontal compression by a factor of 3 and a vertical stretch by a factor of 4.- New x-coordinate: \(x' = x / 3 = -12 / 3 = -4\)- New y-coordinate: \(y' = 4y = 4 \times 18 = 72\)Thus, the image point is (-4, 72).

Transformation for (c)

The transformation is given by: \(y=-2 f(x-6)+4\). This involves shifting the graph 6 units to the right, reflecting over the x-axis, vertically stretching by a factor of 2, and then shifting up by 4 units.- New x-coordinate: \(x' = x + 6 = -12 + 6 = -6\)- New y-coordinate: \(y' = -2y + 4 = -2 \times 18 + 4 = -36 + 4 = -32\)Thus, the image point is (-6, -32).

Transformation for (d)

The transformation is given by: \(y=-2 f\bigg(-\frac{2}{3} x-6\bigg)+4\). This involves reflecting the graph horizontally, horizontally stretching by 3/2, shifting 6 units to the left, reflecting over the x-axis, vertically stretching by a factor of 2, and then shifting up by 4 units.- New x-coordinate: \(x' = -\frac{3}{2}(x + 6) = -\frac{3}{2}(-12 + 6) = -\frac{3}{2}(-6) = 9\)- New y-coordinate: \(y' = -2y + 4 = -2 \times 18 + 4 = -36 + 4 = -32\)Thus, the image point is (9, -32).

Transformation for (e)

The transformation is given by: \(y+3=-\frac{1}{3} f(2(x+6))\). This involves shifting 6 units to the left, horizontally compressing by a factor of 1/2, reflecting over the x-axis, vertically compressing by a factor of 1/3, and then shifting 3 units down.- New x-coordinate: \(x' = \frac{x + 6}{2} = \frac{-12 + 6}{2} = -3\)- New y-coordinate: \(y' = -\frac{1}{3} y - 3 = -\frac{1}{3} \times 18 - 3 = -6 - 3 = -9\)Thus, the image point is (-3, -9).

Key concepts

Precalculus transformations

In precalculus, transformations are a key concept you need to understand for graphing functions. Transformations change the position or shape of a graph. They include translations (shifts), reflections (flips), stretches, and compressions. Knowing how to apply these transformations helps you predict how the graph of a function will look after changes.

• Translations: This involves moving the graph without changing its shape. Horizontal translations shift the graph left or right, while vertical translations move it up or down.


• Reflections: This involves flipping the graph over a specific axis. Reflecting over the x-axis changes the sign of the y-coordinates, and reflecting over the y-axis changes the sign of the x-coordinates.


• Stretches and Compressions: These transformations change the shape of the graph. Vertical stretches/compressions modify the y-coordinates by a factor, while horizontal ones change the x-coordinates.

Mastering these basic transformations will give you a solid foundation for graphing more complex functions in precalculus.

Function transformations

Function transformations are about modifying the graph of a base function through various operations. These operations change the graph's appearance in systematic ways. Here are some common types of transformations:

• Shifts: Moving the entire graph horizontally or vertically. For example, if we have a function f(x), then f(x−h) translates it horizontally by h units, and f(x) + k translates it vertically by k units.


• Reflections: Flipping the graph. For example, −f(x) flips it over the x-axis, and f(−x) flips it over the y-axis.


• Stretches and Compressions: Stretching or compressing the graph. For instance, cf(x) where c > 1 vertically stretches it, and 0 < c < 1 compresses it.

Combining these transformations allows you to manipulate functions and predict their behavior under different changes. This understanding will support your work in more advanced mathematics.

Graph shifts

Graph shifts, also known as translations, are one of the simplest types of graph transformations. They involve moving the entire graph of a function horizontally or vertically without altering its shape. Here's how they work:

• Horizontal Shifts: To shift the graph of a function left or right, you add or subtract a constant to the x-variable. For example, if you have the function f(x), then f(x−h) shifts it to the right by h units and f(x+h) shifts it to the left by h units.


• Vertical Shifts: To shift the graph up or down, you add or subtract a constant to the function f(x) directly. For instance, f(x) + k shifts the graph up by k units, and f(x) − k shifts it down by k units.

Horizontal and vertical shifts are easy to visualize and apply. They are fundamental to understanding more complex graph transformations in calculus and beyond.