Linear Functions Overview

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Questions and Answers

What is the value of f(9) for the function f(x) = x + 4, and what point does this correspond to on the graph?

f(9) = 13, corresponding to the point (9, 13) on the graph.

How can you determine the average rate of change for a linear function between two x-values?

The average rate of change is determined by calculating the difference in y-values divided by the difference in x-values.

Identify the y-intercept of the function y = -x + 7 and explain its significance.

The y-intercept is 7, which corresponds to the point (0, 7) on the graph where the line crosses the y-axis.

What is the slope of the linear function y = -x + 7, and how does this affect the behavior of the function?

<p>The slope is -1, indicating that the function is decreasing as x increases.</p> Signup and view all the answers

Explain how to graph a linear function without creating a table of values.

<p>To graph a linear function, use the slope and y-intercept to plot the line directly.</p> Signup and view all the answers

Flashcards

Linear Function

A function whose graph is a straight line.

Slope of a line

The rate of change of y with respect to x; the average rate of change

y-intercept

The point where the graph of a function crosses the y-axis

Equation of a line

y = mx + b, where m is the slope and b is the y-intercept.

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Average rate of change

The change in y divided by the change in x between two points on a graph.

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Study Notes

Linear Functions

  • Linear functions are functions whose graphs are straight lines.
  • A linear function is in the form f(x) = mx + b or y = mx + b, where m is the slope and b is the y-intercept.

Exercise #1

  • The function f(x) = (2/3)x + 4 is a linear function.
  • Finding f(9): f(9) = (2/3)(9) + 4 = 6 + 4 = 10. The point (9, 10) lies on the graph.
  • A table of values for the function are included, plotting these points gives a straight line.
  • The average rate of change between any two consecutive x-values is equal to the slope of the line. This value should be 2/3
  • The y-intercept is the point where the line crosses the y-axis. It's the value of y when x = 0. In this case, the y-intercept is 4.

Linear Functions Definition

  • A linear function has the form f(x) = mx + b (or y = mx + b)
  • m represents the slope of the line.
  • b represents the y-intercept, the point where the line crosses the y-axis.

Exercise #2

  • For the function y = 5x + 2, choose two x-values, calculate their corresponding y-values, and then calculate the average rate of change.
  • The average rate of change is equal to the slope of the line (m = 5).

Exercise #3 (Graphing a Linear Function)

  • The given linear function is y = -x/2 + 7
  • To graph, create a table of (x,y) values.
  • Plot the points on the graph and draw a straight line through them.
  • The slope of the line is -1/2
  • The line is decreasing because the slope is negative.

Exercise #4 (Graphing Linear Equations)

  • Provides several equations (y = 2x - 3, y = -x + 6 etc.)
  • Find the slope and y-intercept for each equation.
  • Graph these equations on a grid.

Exercise #1 (Multiple Choice Lines)

  • Four lines are graphed on a coordinate plane.
  • Match each graph to its correct equation.

Exercise #2 (Graphing and Identifying)

  • Graph the equation y = (-3/2)x + 4 .
  • Identify the slope and the y-intercept.

Exercise #3 (Average Rate of Change)

  • Find two sets of coordinate points on the line.
  • Calculate the average rate of change between the two coordinate pairs.
  • The results is equal to the slope of the line.

Exercise #4 (Rewriting equations)

  • The equation 2y - 6x = 12 is given.
  • Steps are shown in the text to rewrite the equation in the slope-intercept form (y = mx + b).
  • Identify the slope and y-intercept of the rewritten equation.

Exercise #5 (Rewriting equations)

  • Several equations in various forms (e.g. 3y - 3x = 15) are provided
  • Rewrite the equations in the slope-intercept form (y = mx + b)
  • Identify the slope and y-intercept for each rewritten equation.

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