Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 47

Chapter 3, Problem 47

The graph of a linear function f is shown. (a) Identify the slope, y-intercept, and x-intercept. (b) Write an equation that defines f.

Verified step by step guidance

Step 1: Identify the y-intercept by finding the point where the line crosses the y-axis. This occurs when \(x=0\). From the graph, observe the y-coordinate of this point.

Step 2: Identify the x-intercept by finding the point where the line crosses the x-axis. This occurs when \(y=0\). From the graph, observe the x-coordinate of this point.

Step 3: Calculate the slope \(m\) of the line using the formula \(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. Use the intercepts or any two points clearly visible on the graph.

Step 4: Write the equation of the line in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope found in Step 3 and \(b\) is the y-intercept found in Step 1.

Step 5: Verify the equation by plugging in the x-values of the intercepts or other points on the line to check if the corresponding y-values satisfy the equation.

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Key Concepts

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

Slope of a Linear Function

The slope measures the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points. It indicates how much y changes for a unit change in x. A positive slope means the line rises from left to right, while a negative slope means it falls.

Intercepts of a Linear Function

The y-intercept is the point where the line crosses the y-axis (x=0), representing the output value when the input is zero. The x-intercept is where the line crosses the x-axis (y=0), showing the input value that makes the output zero. Identifying these points helps in graphing and writing the equation of the line.

Equation of a Linear Function

A linear function can be expressed in the form y = mx + b, where m is the slope and b is the y-intercept. This equation defines the relationship between x and y, allowing you to calculate y for any x. Writing the equation requires knowing the slope and y-intercept from the graph.

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