Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 41

3장, 문제 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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 specific 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 allows for a clearer understanding of the line's behavior.

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 positive slope indicates the line rises as it moves from left to right, while a negative slope indicates it falls. In this question, the slope is given as -6, meaning for every unit increase in x, y decreases by 6 units, which is crucial for forming the line's equation.