Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problema 112

Capitolo 4, Problema 112

Write an equation in point-slope form and slope-intercept form of the line passing through (-10,3) and (-2,-5).

Guida verificata passo dopo passo

First, find the slope (m) of the line passing through the points (-10, 3) and (-2, -5) using the slope formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substitute the values:

m = \frac{-5 - 3}{-2 - (-10)}

Simplify the numerator and denominator to find the slope:

m = \frac{-8}{8}

. Then simplify the fraction to get the slope value.

Use the point-slope form of a line equation:

y - y_1 = m(x - x_1)

. Choose one of the given points, for example (-10, 3), and substitute

m

x_1 = -10

, and

y_1 = 3

into the equation.

Write the point-slope form equation explicitly with the substituted values:

y - 3 = m(x + 10)

To convert to slope-intercept form, solve the point-slope equation for

y

by distributing the slope and isolating

y

on one side, resulting in an equation of the form

y = mx + b

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Slope of a Line

The slope measures the steepness of a line and is calculated as the change in y-values divided by the change in x-values between two points. For points (x₁, y₁) and (x₂, y₂), slope m = (y₂ - y₁) / (x₂ - x₁). It indicates how much y changes for a unit change in x.

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The Slope of a Line

Point-Slope Form of a Line

Point-slope form expresses a line's equation using a known point (x₁, y₁) and the slope m: y - y₁ = m(x - x₁). This form is useful for writing the equation quickly when a point and slope are known, directly relating changes in x and y.

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Point-Slope Form

Slope-Intercept Form of a Line

Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. It clearly shows the slope and where the line crosses the y-axis, making it easy to graph and understand the line's behavior.

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Graphing Lines in Slope-Intercept Form

Pratica correlata

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Exercises 107–109 will help you prepare for the material covered in the next section. Factor: x³+3x²−x−3

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Domanda del libro di testo

Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x⁴−2x²+1 is even, odd, or neither. Describe the symmetry, if any, for the graph of f.

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Domanda del libro di testo

Solve: $x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4 = 0$ .