Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 11

Chapter 3, Problem 11

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = 2, passing through (3, 5)

Verified step by step guidance

Recall the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line.

Substitute the given slope \(m = 2\) and the point \((3, 5)\) into the point-slope form: \(y - 5 = 2(x - 3)\).

To convert to slope-intercept form, start by distributing the slope on the right side: \(y - 5 = 2x - 6\).

Next, isolate \(y\) by adding 5 to both sides: \(y = 2x - 6 + 5\).

Simplify the right side to get the slope-intercept form: \(y = 2x - 1\).

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Key Concepts

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

Point-Slope Form of a Line

The point-slope form is an equation of a line expressed as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. It is useful for writing the equation when you know a point and the slope.

Slope-Intercept Form of a Line

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This form clearly shows the slope and where the line crosses the y-axis, making it easy to graph and interpret.

Using Given Conditions to Find the Equation

Given a slope and a point, you substitute these values into the point-slope form to write the equation. Then, by simplifying and solving for y, you convert it into slope-intercept form, revealing the y-intercept.

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Evaluate each function at the given values of the independent variable and simplify. (a) f(-2), (b) f(1), (c) f(2)

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Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = ∛(x − 4) and g(x) = x³ +4

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Use the graph of y = f(x) to graph each function g.

g(x) = f(-x)+3

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Textbook Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (4, −7) and perpendicular to the line whose equation is x − 2y – 3 = 0

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The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = x +3

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Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (3.5, 8.2) and (-0.5, 6.2)

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