Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 59

Chapter 3, Problem 59

Graph the line passing through the given point and having the indicated slope. Plot two points on the line. through (3, -4), m = - 1/3

Verified step by step guidance

Identify the given point and slope. The point is (3, -4) and the slope is $ m = -\frac{1}{3} $.

Use the point-slope form of the equation of a line: $ y - y_1 = m(x - x_1) $, where $ (x_1, y_1) $ is the given point.

Substitute the given point and slope into the point-slope form: $ y - (-4) = -\frac{1}{3}(x - 3) $. Simplify this to get $ y + 4 = -\frac{1}{3}(x - 3) $.

Find the y-coordinate of a second point by choosing a value for $ x $ different from 3, substitute it into the equation, and solve for $ y $. This gives you a second point on the line.

Use the slope $ m = -\frac{1}{3} $ to find a third point by moving 3 units horizontally and 1 unit vertically from the original point, following the slope direction. Plot the original point and these two points, then draw the line through them.

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Key Concepts

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

Slope of a Line

The slope represents the rate of change or steepness of a line, calculated as the ratio of the vertical change to the horizontal change between two points. A slope of -1/3 means the line falls 1 unit vertically for every 3 units it moves horizontally to the right.

Point-Slope Form of a Line

The point-slope form, y - y₁ = m(x - x₁), is used to write the equation of a line when a point (x₁, y₁) and slope m are known. It helps in finding other points on the line by substituting values for x or y.

Plotting Points on a Coordinate Plane

Plotting points involves marking specific (x, y) coordinates on the Cartesian plane. To graph a line, you plot the given point and use the slope to find a second point by moving horizontally and vertically according to the slope.

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Related Practice

Textbook Question

Determine whether each function is even, odd, or neither. ƒ(x)=0.5x⁴-2x²+6

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Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(k)

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Solve each problem. See Example 4. Suppose that the cost of mailing a letter weighing x ounces, where x>0, is ƒ(x)=55-15[[1-x]]cents. What is the cost for the first ounce?

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Textbook Question

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5.

$(ƒ \circ g) (- 2)$

Textbook Question

For each line described, write an equation in (a) slope-intercept form, if possible, and (b) standard form. through (3, -5) with slope -2.

Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x²+4x+1. Find each of the following. Simplify if necessary. ƒ(p)

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