Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 14a

3장, 문제 14a

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = 8, passing through (4, −1)

검증된 단계별 안내

Step 1: Recall 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.

Step 2: Substitute the given slope

m = 8

and the point

(x_1, y_1) = (4, -1)

into the point-slope form. This gives:

y - (-1) = 8(x - 4)

Step 3: Simplify the equation from Step 2. The double negative becomes positive, so the equation becomes:

y + 1 = 8(x - 4)

. This is the equation in point-slope form.

Step 4: To convert to slope-intercept form, expand the equation

y + 1 = 8(x - 4)

. Distribute the

8

to get:

y + 1 = 8x - 32

Step 5: Isolate

y

by subtracting

1

from both sides of the equation. This gives:

y = 8x - 33

. This is the equation in slope-intercept form.

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주요 개념

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

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, allowing for straightforward calculations and graphing.

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 the point where the line crosses the y-axis, making it easier to graph the line and understand its behavior.

Slope

Slope is a measure of the steepness or incline of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this case, a slope of 8 indicates that for every unit increase in x, y increases by 8 units, which significantly influences the line's angle and position.