If one point on a line is (2, −6) and the line's slope is -3/2, find the y-intercept.

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Recall the slope-intercept form of a line: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Substitute the given slope \(m = -\frac{3}{2}\) into the equation: \(y = -\frac{3}{2}x + b\).

Use the given point \((2, -6)\) by substituting \(x = 2\) and \(y = -6\) into the equation: \(-6 = -\frac{3}{2} \times 2 + b\).

Simplify the multiplication on the right side: \(-6 = -3 + b\).

Solve for \(b\) by adding 3 to both sides: \(b = -6 + 3\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Slope of a Line

The slope measures the steepness and direction of a line, defined as the ratio of the change in y to the change in x between two points. It is often represented as 'm' and is crucial for writing the equation of a line.

Slope-Intercept Form of a Line

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This form allows easy identification of the line's slope and where it crosses the y-axis, facilitating the calculation of unknown parameters.

Using a Point to Find the Y-Intercept

Given a point (x, y) on the line and the slope m, substitute these values into y = mx + b to solve for b, the y-intercept. This method uses known information to find the line's vertical intercept.

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Textbook Question

In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x+3|

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Textbook Question

Use the graphs of f and g to solve Exercises 83–90.

Find (f+g)(−3).