Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 20a

3장, 문제 20a

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = −1, passing through (−4, − 1/4)

검증된 단계별 안내

Step 1: Recall the point-slope form of a linear equation, which is given by:

y - y

₁ = m(x - x₁), where

m

is the slope and

(x

₁, y₁) is a point on the line.

Step 2: Substitute the given slope

m = -1

and the point

(x

₁, y₁) = (-4, -1/4) into the point-slope form. This gives:

y - (-1/4) = -1(x - (-4))

Step 3: Simplify the equation from Step 2. First, simplify the double negatives:

y + 1/4 = -1(x + 4)

Step 4: Expand the equation to convert it into slope-intercept form

y = mx + b

. Distribute the slope

-1

on the right-hand side:

y + 1/4 = -x - 4

Step 5: Isolate

y

by subtracting

1/4

from both sides:

y = -x - 4 - 1/4

. Combine like terms to simplify the constant term.

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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 given a point on the line and its slope. The formula is written 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 quickly writing the equation of a line when you know a point and the slope.

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 over the change in x (rise over run). A slope of -1 indicates that for every unit increase in x, y decreases by one unit, resulting in a downward slant. Understanding slope is crucial for graphing lines and interpreting their behavior in relation to the coordinate plane.