91Ó°ÊÓ

Which one of the following is true? a. If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of three units to the right and three units upward of the graph of \(f\) b. If \(f(x)=-\sqrt{x}\) and \(g(x)=\sqrt{-x},\) then \(f\) and \(g\) have identical graphs. c. If \(f(x)=x^{2}\) and \(g(x)=5\left(x^{2}-2\right),\) then the graph of \(g\) can be obtained from the graph of \(f\) by stretching \(f\) five units followed by a downward shift of two units. d. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting in the \(x\) -axis, and then moving the resulting graph down four units.

Step by step solution

01

Statement Evaluation 1

Examine statement a. The given pair of functions are \(f(x) = |x|\) and \(g(x) = |x + 3| + 3\). A positive horizontal shift would result in the absolute value function becoming \(|x - c|\), where c is the shift factor, not \(|x + c|\). Therefore, the statement is false because the graph of function \(g(x) = |x + 3| + 3\) is a translation of the graph of \(f(x) = |x|\) three units to the left and three units upward.

02

Statement Evaluation 2

Next, look at statement b. The pair of functions here are \(f(x) = -\sqrt{x}\) and \(g(x) = \sqrt{-x}\). Plotting these functions, we see that the transformation is not just a reflection about the x-axis. Therefore, the statement is false, as \(g(x) = \sqrt{-x}\) rotates \(f(x) = -\sqrt{x}\) by 90 degrees anticlockwise about the origin, instead of creating an identical graph.

03

Statement Evaluation 3

The third statement c is about functions \(f(x) = x^{2}\) and \(g(x) = 5(x^{2} - 2)\). This statement claims that \(g\) is obtained by stretching \(f\) by 5 units vertically and then shifting it downwards by 2 units. Instead, the transformation from \(f\) to \(g\) involves a vertical stretch by a factor of 5 followed by a vertical shift downwards by 10 units (\(5 * -2\)). So, the statement is not true.

04

Statement Evaluation 4

Finally, look at statement d that involves functions \(f(x) = x^{3}\) and \(g(x) = -(x - 3)^{3} - 4\). Here, \(g\) is a transformation of \(f\) that involves a horizontal shift of 3 units to the right, a reflection about the x-axis, and a vertical shift of 4 units downwards. So this statement is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function

An absolute value function is graphically represented by a V-shaped curve. It reflects the magnitude of a number, disregarding its sign, making all outputs non-negative. In mathematical terms, for any real number input x, the absolute value function is defined as \( f(x) = |x| \) which yields x if x is positive or zero, and -x if x is negative.

The absolute value graph has a corner point at the origin \( (0, 0) \) where the direction of the graph changes. This characteristic point is called the vertex of the absolute value graph. Any transformation such as a shift, reflection, or stretch, starts from this baseline graph.

Graph Translation

Graph translation involves sliding the entire graph of a function horizontally or vertically without altering its shape or orientation. A vertical translation shifts the graph up or down, while a horizontal translation moves the graph left or right.

Specifically, for a function \( f(x) \) if we add a constant 'c' to the function to get \( f(x) + c \) it results in a vertical shift of 'c' units upwards if c is positive, and 'c' units downwards if c is negative. Likewise, \( f(x - c) \) will translate the graph 'c' units to the right, while \( f(x + c) \) moves it 'c' units to the left.

Graph Reflection

Graph reflection flips the graph of a function over a specific line, such as the x-axis or y-axis, altering the graph's orientation but not its shape. A reflection over the x-axis can be achieved by multiplying the function by -1, changing its sign, \( f(x) \) to \( -f(x) \). This transformation mirrors every point of the graph across the x-axis.

For a reflection over the y-axis, replace x with -x in the function, transforming \( f(x) \) into \( f(-x) \). This causes the graph to flip horizontally, mirroring it over the y-axis. It's important to visualize these reflections to fully grasp their impact on the graph's shape.

Vertical Stretch

A vertical stretch scales the graph of a function away from the x-axis, either elongating or compressing it vertically. This is done by multiplying the function by a factor greater than 1 for a stretch or between 0 and 1 for a compression.

When a function \( f(x) \) is transformed into \( a \cdot f(x) \) with \( a > 1 \) there is a vertical stretch by a factor of 'a'. This multiplication increases the distance of each point from the x-axis proportionally, broadening the graph vertically while keeping the x-coordinates unchanged.

Horizontal Shift

A horizontal shift, similar to a horizontal translation, moves the graph of a function to the left or right along the x-axis. This is achieved by adding or subtracting a constant from the input variable x.

To move the graph to the right by 'h' units, we modify the function from \( f(x) \) to \( f(x - h) \) and to shift it left by 'h' units, we use \( f(x + h) \) instead. This change affects all the x-coordinates of the graph's points while maintaining their corresponding y-coordinates and the overall shape of the graph.