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Chapter 3: Problem 32

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift to the left 1 unit, stretch vertically by a factor of \(3,\) and shift upward 10 units

Short Answer

Expert verified
The final equation is \(k(x) = 3|x + 1| + 10\).

Step by step solution

01

Initial Function

We start with the given function, which is the absolute value function: \( f(x) = |x| \). This is a V-shaped graph centered at the origin.
02

Horizontal Shift

The first transformation is a shift to the left by 1 unit. To shift a function to the left by \(1\) unit, we apply the transformation \(x' = x + 1\). So the function becomes \( g(x) = |x + 1| \).
03

Vertical Stretch

Next, we apply a vertical stretch by a factor of \(3\). To vertically stretch a function, we multiply it by \(3\). Thus, the new function becomes \( h(x) = 3|x + 1| \).
04

Vertical Shift

Finally, we shift the graph upward by \(10\) units. This involves adding \(10\) to the function, giving us \( k(x) = 3|x + 1| + 10 \).
05

Final Transformed Function

After performing all the transformations, the final equation of the transformed graph is \( k(x) = 3|x + 1| + 10 \).

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Key Concepts

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

Absolute Value Function
The absolute value function is a fundamental function in mathematics, denoted as \( f(x) = |x| \). It produces the non-negative value of \( x \), reflecting any negative input across the \( y \)-axis to make it positive. The graph of an absolute value function creates a V-shape, with its vertex typically located at the origin \((0,0)\). This characteristic V-shape makes the absolute value function easy to identify visually in graphs.

The basic properties of the absolute value function include:
  • The domain is all real numbers \((-\infty, \infty)\).
  • The range is all non-negative real numbers \([0, \infty)\).
  • The function is symmetric about the y-axis, demonstrating even function properties.
  • There is a sharp corner at the vertex, giving it the distinctive V shape.
This function is widely used in various fields, including physics, engineering, and economics, where the non-negativity of values is crucial.
Horizontal Shift
A horizontal shift involves moving the graph of a function left or right across the \( x \)-axis. For the function \( f(x) = |x| \), shifting it horizontally means changing the structure inside the absolute value brackets.

To shift the function \( f(x) = |x| \) to the left by 1 unit, we replace \( x \) with \( x + 1 \). This transformation results in a new function: \( g(x) = |x + 1| \). The notation \( x + 1 \) indicates that every point on the graph moves leftward by 1 unit.

Important points to understand about horizontal shifts:
  • \( f(x + c) \) means shifting the function to the left by \( c \) units.
  • \( f(x - c) \) moves the graph to the right by \( c \) units.
Horizontal shifts do not alter the shape of the graph; they only displace it laterally across the grid.
Vertical Stretch
A vertical stretch involves pulling the graph of a function away from the x-axis, effectively stretching it vertically. This is achieved by multiplying the function by a constant factor. For the transformed function \( g(x) = |x + 1| \) from the previous step, multiplying by 3 results in \( h(x) = 3|x + 1| \).

The transformation rule for vertical stretching is straightforward:
  • Multiply the entire function by a positive constant \( a \). If \( a > 1 \), the graph stretches away from the x-axis.
  • If \( 0 < a < 1 \), the graph compresses toward the x-axis, but this scenario deals specifically with stretching.
This vertical stretch increases the "height" of the V-shaped graph without altering its overall direction or symmetry. Each point's distance from the x-axis is multiplied by the stretching factor, making the graph appear taller.
Vertical Shift
In a vertical shift, the entire graph of a function moves up or down the y-axis. For our function \( h(x) = 3|x + 1| \), shifting it 10 units upward involves adding 10 to the entire function, resulting in \( k(x) = 3|x + 1| + 10 \).

This transformation has specific characteristics:
  • Adding a positive constant shifts the graph upward.
  • Subtracting a constant moves the graph downward.
Vertical shifts do not change the shape or horizontal position of the graph; they only elevate or lower every point by the same number of units.

In this final form, \( k(x) = 3|x + 1| + 10 \) reflects all the transformations combined: a leftward shift, a vertical stretch, and an upward shift, yielding the final position and shape of the graph.

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Most popular questions from this chapter

Highway Engineering A highway engineer wants to estimate the maximum number of cars that can safely travel a particular highway at a given speed. She assumes that each car is 17 \(\mathrm{ft}\) long, travels at a speed \(s,\) and follows the car in front of it at the "safe following distance" for that speed. She finds that the number \(N\) of cars that can pass a given point per minute is modeled by the function $$ N(s)=\frac{88 s}{17+17\left(\frac{s}{20}\right)^{2}} $$ At what speed can the greatest number of cars travel the highway safely?

If \(f(x)=\sqrt{2 x-x^{2}}\) graph the following functions in the viewing rectangle \([-5,5]\) by \([-4,4]\) , How is each graph related to the graph in part (a)? \(\begin{array}{llll}{\text { (a) } y=f(x)} & {\text { (b) } y=f(- x)} & {\text { (c) } y=-f\left(-x\right)}\\\ {\text { (d) } y=f(- 2x)} & {\text { (e) } y=f\left(-\frac{1}{2} x\right)}\end{array}\)

Multiple Discounts You have a \(\$ 50\) coupon from the manufacturer good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a 20\(\%\) discount on all cell phones. Let \(x\) represent the regular price of the cell phone. (a) Suppose only the 20\(\%\) discount applies. Find a function \(f\) that models the purchase price of the cell phone as a function of the regular price \(x .\) (b) Suppose only the \(\$ 50\) coupon applies. Find a function \(g\) that models the purchase price of the cell phone as a function of the sticker price \(x .\) (c) If you can use the coupon and the discount, then the purchase price is either \(f \circ g(x)\) or \(g\) o \(f(x),\) depending on the order in which they are applied to the price. Find both \(f \circ g(x)\) and \(g \circ f(x) .\) Which composition gives the lower price?

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=\sqrt{x}, \quad g(x)=\frac{x}{x-1}, \quad h(x)=\sqrt[3]{x} $$

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=\frac{1}{x}, \quad g(x)=x^{3}, \quad h(x)=x^{2}+2 $$
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