Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 41

Chapter 3, Problem 41

Graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = -2x, g(x) = -2x-1

Verified step by step guidance

Identify the given functions: $f(x) = -2x$ and $g(x) = -2x - 1$. Notice that both are linear functions with the same slope but different y-intercepts.

Create a table of values for $x$ starting from $-2$ to $2$ for both functions. For each $x$, calculate $f(x)$ using $f(x) = -2x$ and $g(x)$ using $g(x) = -2x - 1$.

Plot the points for $f(x)$ on the coordinate plane by marking the pairs $(x, f(x))$ for $x = -2, -1, 0, 1, 2$. Connect these points to form the graph of $f$.

Similarly, plot the points for $g(x)$ by marking the pairs $(x, g(x))$ for the same $x$ values and connect these points to form the graph of $g$.

Compare the two graphs: since $g(x)$ differs from $f(x)$ by a constant $-1$, the graph of $g$ is a vertical shift of the graph of $f$ downward by 1 unit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Notation and Evaluation

Function notation, such as f(x) and g(x), represents a rule that assigns each input x to an output value. Evaluating a function involves substituting specific x-values into the function's formula to find corresponding y-values, which are essential for plotting points on a graph.

Graphing Linear Functions

Linear functions produce straight-line graphs characterized by a constant rate of change or slope. Plotting points for selected x-values and connecting them helps visualize the function, revealing properties like slope and y-intercept that define the line's position and direction.

Transformations of Graphs (Vertical Shifts)

A vertical shift moves a graph up or down without changing its shape. In the given functions, g(x) = f(x) - 1 represents a downward shift of the graph of f(x) by 1 unit, meaning every point on f's graph is moved one unit lower to obtain g's graph.

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Related Practice

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In Exercises 41–44, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-3, 2) with slope - 6

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Textbook Question

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$f$\left$(x$\right$)=$\sqrt{x}$$ , $g$\left$(x$\right$)=x-5$

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Textbook Question

In Exercises 39–48, give the slope and y-intercept of each line whose equation is given. Then graph the linear function. f(x) = -2x+1

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