Holt McDougal Algebra Graphing Linear Functions Toolbox 2.3 (a)

Holt McDougal Algebra Graphing Linear Functions How do you determine whether a function is linear? How do you graph a linear function given two points, a table, an equation, or a point and a slope? Essential Questions

Holt McDougal Algebra Graphing Linear Functions Time (h)01234 Distance from Land (mi) –25 +1 –25 +1 –25 +1 –25 This rate can be expressed as. Notice that the rate of change is constant. The hurricane moves 25 miles closer each hour.

Holt McDougal Algebra Graphing Linear Functions Functions with a constant rate of change are called linear functions. A linear function can be written in the form f(x) = mx + b, where x is the independent variable and m and b are constants. The graph of a linear function is a straight line made up of all points that satisfy y = f(x).

Holt McDougal Algebra Graphing Linear Functions Determine whether the data set could represent a linear function. Example 1A: Recognizing Linear Functions x–2024 f(x)f(x)210–1 +2 –1 +2 –1 +2 –1 The rate of change,, is constant. So the data set is linear.

Holt McDougal Algebra Graphing Linear Functions Determine whether the data set could represent a linear function. Example 1B: Recognizing Linear Functions x2345 f(x)f(x) The rate of change,, is not constant. 2 ≠ 4 ≠ 8. So the data set is not linear.

Holt McDougal Algebra Graphing Linear Functions The constant rate of change for a linear function is its slope. The slope of a linear function is the ratio, or. The slope of a line is the same between any two points on the line. You can graph lines by using the slope and a point.

Holt McDougal Algebra Graphing Linear Functions Example 2: Graphing Lines Using Slope and a Point Plot the point (–1, –3). Graph the line with slope that passes through (–1, –3). The slope indicates a rise of 5 and a run of 2. Move up 5 and right 2 to find another point. Then draw a line through the points.

Holt McDougal Algebra Graphing Linear Functions Linear functions can also be expressed as linear equations of the form y = mx + b. When a linear function is written in the form y = mx + b, the function is said to be in slope-intercept form because m is the slope of the graph and b is the y-intercept. Notice that slope-intercept form is the equation solved for y.

Holt McDougal Algebra Graphing Linear Functions Example 3A: Graphing Functions in Slope-Intercept Form Solve for y first. Write the function –4x + y = –1 in slope-intercept form. Then graph the function. –4x + y = –1 y = 4x – 1 Add 4x to both sides. +4x The line has y-intercept –1 and slope 4, which is. Plot the point (0, –1). Then move up 4 and right 1 to find other points.

Holt McDougal Algebra Graphing Linear Functions Example 3B: Graphing Functions in Slope-Intercept Form Solve for y first. Write the function in slope-intercept form. Then graph the function. Distribute. The line has y-intercept 8 and slope. Plot the point (0, 8). Then move down 4 and right 3 to find other points. Multiply both sides by

Holt McDougal Algebra Graphing Linear Functions An equation with only one variable can be represented by either a vertical or a horizontal line.

Holt McDougal Algebra Graphing Linear Functions Vertical and Horizontal Lines Vertical LinesHorizontal Lines The line x = a is a vertical line at a. The line y = b is a horizontal line at b.

Holt McDougal Algebra Graphing Linear Functions The slope of a vertical line is undefined. The slope of a horizontal line is zero.

Holt McDougal Algebra Graphing Linear Functions Example 4: Graphing Vertical and Horizontal Lines Determine if each line is vertical or horizontal. A. x = 2 B. y = –4 This is a vertical line located at the x-value 2. (Note that it is not a function.) This is a horizontal line located at the y-value –4. x = 2 y = –4