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Chapter 1: Problem 100

How do the graphs of \(f(x)\) and \(f(x+10)+10\) differ?

Short Answer

Expert verified
The graph of \(f(x+10)+10\) shifts left 10 units and up 10 units compared to \(f(x)\).

Step by step solution

01

Understand the Graph Transformation

The function \(f(x)\) represents some graph on a coordinate plane. We are dealing with a modification of this function: \(f(x+10)+10\). This modification involves transformations that can affect the graph's position on the plane.
02

Horizontal Shift Interpretation

The expression \(f(x+10)\) implies a horizontal shift. The \(+10\) inside the parentheses indicates a shift to the left by 10 units. In general, \(f(x+c)\) shifts the graph \(c\) units left if \(c > 0\).
03

Vertical Shift Interpretation

The outer \(+10\) in \(f(x+10)+10\) signifies a vertical shift. This increases each output value of the function by 10, effectively moving the entire graph upward by 10 units on the coordinate plane.
04

Determine the Overall Transformation

Considering both transformations together, the graph of \(f(x+10)+10\) is obtained by shifting the original graph \(f(x)\) left by 10 units and then up by 10 units.

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Key Concepts

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

Horizontal Shift
A horizontal shift involves moving the graph of a function left or right on the coordinate plane. Understanding how this affects the graph can greatly help in visual development.
  • The expression inside the function, such as \(f(x+c)\), indicates a horizontal shift.
  • If \(c\) is a positive number, the shift is to the left by \(c\) units. So, in the function \(f(x+10)\), the graph moves 10 units to the left.
  • If \(c\) were negative, the shift would be to the right by \(-c\) units.
Horizontal shifts happen because altering the input values affects the entire graph's position along the x-axis. It's like sliding the graph horizontally without changing its shape.
Vertical Shift
Vertical shifts modify the position of a graph up or down along the y-axis on the coordinate plane. This type of transformation is intuitive as it deals directly with the output values.
  • For a vertical shift, look at terms added or subtracted outside of the function form, such as \(f(x) + k\).
  • If \(k\) is positive, the graph moves up by \(k\) units. Hence, \(f(x+10) + 10\) raises the graph 10 units up.
  • If \(k\) is negative, it moves the graph down by \(-k\) units.
Vertical shifts occur because the adjustment changes every output, or y-value, of the function. This does not alter the graph's horizontal features or wideness.
Coordinate Plane
The coordinate plane is a two-dimensional space where graphs live, using perpendicular axes for orientation.
  • The horizontal line is the x-axis, which captures horizontal movements like shifts or stretches.
  • The vertical line is the y-axis, designed for vertical changes in position or elongation.
  • The intersection of these axes is the origin, denoted as (0,0), and is the center of all transformations.
Knowing how the coordinate plane works enhances comprehension of how transformations visually alter graphs. It serves as a map where every point on a graph can be accurately located based on its x (horizontal position) and y (vertical position) values.

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Most popular questions from this chapter

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ f(x)=7 x^{2}-3 x+2 $$

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ \begin{array}{l} f(x)=x^{3} \\ \text { [Hint: Use } \left.(x+h)^{3}=x^{3}+3 x^{2} h+3 x h^{2}+h^{3} .\right] \end{array} $$

For each function, find and simplify \(f(x+h)\). $$ f(x)=2 x^{2}-5 x+1 $$

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ f(x)=\frac{1}{x^{2}} $$

For each pair of functions \(f(x)\) and \(g(x)\), find and fully simplify a. \(f(g(x))\) and b. \(g(f(x))\) $$ f(x)=2 x-6 ; \quad g(x)=\frac{x}{2}+3 $$
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