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

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

Verified step by step guidance

Identify the point-slope form of a linear equation, which is given by:

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 = -6

and the point

(x_1, y_1) = (-3, 2)

into the point-slope form. This gives:

y - 2 = -6(x - (-3))

Simplify the equation from step 2 to get the point-slope form:

y - 2 = -6(x + 3)

To convert to slope-intercept form, expand the equation from step 3. Distribute

-6

across

(x + 3)

, resulting in:

y - 2 = -6x - 18

Solve for

y

by adding

2

to both sides of the equation:

y = -6x - 16

. This is the slope-intercept form of the equation.

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Key Concepts

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

Point-Slope Form

The point-slope form of a linear equation is expressed as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is particularly useful for writing equations when you know a point on the line and the slope. In this case, with the point (-3, 2) and a slope of -6, you can directly substitute these values into the formula.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope and b is the y-intercept. This form is advantageous for quickly identifying the slope and where the line crosses the y-axis. To convert from point-slope to slope-intercept form, you can rearrange the equation after substituting the known values.

Slope

Slope is a measure of the steepness or incline of a line, calculated as the change in y divided by the change in x (rise over run). A negative slope indicates that as x increases, y decreases, which is the case here with a slope of -6. Understanding slope is crucial for interpreting the direction of the line and for converting between different forms of linear equations.

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