Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 16a

3장, 문제 16a

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = -5, passing through (-4, -2)

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Start with the point-slope form of a linear 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 = -5

and the point

(x_1, y_1) = (-4, -2)

into the point-slope form. This gives:

y - (-2) = -5(x - (-4))

Simplify the equation by removing the double negatives:

y + 2 = -5(x + 4)

. This is the equation in point-slope form.

To convert to slope-intercept form, expand the equation: Distribute

-5

to both terms inside the parentheses:

y + 2 = -5x - 20

Isolate

y

by subtracting

2

from both sides:

y = -5x - 22

. This is the equation in slope-intercept form.

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주요 개념

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Point-Slope Form

Point-slope form is a way to express the equation of a line when you know the slope and a point on the line. It is written as y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope. This form is particularly useful for quickly writing the equation of a line when given a slope and a specific point.

Slope-Intercept Form

Slope-intercept form is another way to express the equation of a line, defined as y = mx + b, where m is the slope and b is the y-intercept. This form is beneficial for easily identifying the slope and where the line crosses the y-axis. Converting from point-slope to slope-intercept form involves rearranging the equation to isolate y.

Slope

The slope of a line measures its steepness and direction, calculated as the change in y divided by the change in x (rise over run). A negative slope indicates that the line descends from left to right, while a positive slope indicates it ascends. Understanding slope is crucial for graphing lines and interpreting their behavior in relation to other lines.