Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 63

Chapter 3, Problem 63

In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 8x – 4y – 12 =0

Verified step by step guidance

Start by rewriting the given equation $8x - 4y - 12 = 0$ in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Isolate the $y$ term on one side. Add \$4y$ to both sides and subtract $12$ from both sides to get $8x - 12 = 4y$.

Divide every term by 4 to solve for $y$: $\frac{8x}{4} - \frac{12}{4} = y$, which simplifies to $y = 2x - 3$.

Identify the slope $m$ and y-intercept $b$ from the equation $y = 2x - 3$. Here, the slope $m$ is 2, and the y-intercept $b$ is $-3$.

To graph the function, start by plotting the y-intercept at $(0, -3)$ on the coordinate plane. Then use the slope $2$ (which means rise over run = 2/1) to find another point by moving up 2 units and right 1 unit from the y-intercept, and draw the line through these points.

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Key Concepts

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

Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. Rewriting an equation into this form makes it easier to identify these values and to graph the line.

Slope of a Line

The slope (m) measures the steepness and direction of a line, calculated as the ratio of the change in y to the change in x (rise over run). It indicates how much y changes for a unit change in x and is essential for understanding the line's behavior.

Graphing Using Slope and Y-Intercept

Graphing a linear function involves plotting the y-intercept (where the line crosses the y-axis) and then using the slope to find other points by moving vertically and horizontally. This method provides a straightforward way to visualize the line on the coordinate plane.

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Graphing Lines in Slope-Intercept Form

Related Practice

Textbook Question

In Exercises 59-64, let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. g (f[h (1)])

Textbook Question

Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. f(g[h (1)])

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Textbook Question

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. r(x) = -(x + 1)²

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Textbook Question

Use the vertical line test to identify graphs in which y is a function of x.

Textbook Question

Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2).

f(x) = 2x-3, g(x) = (x+3)/2

Textbook Question

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (1/2)(x − 1)²

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