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Section 1.2

Section 1.2. Linear Functions and Graphs. EXAMPLE. The following table shows the cost per day for a rental car depending on how many miles you drive a day. LINEAR FUNCTIONS. Definition : A linear function is a function of the form f ( x ) = mx + b.

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Section 1.2

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  1. Section 1.2 Linear Functions and Graphs

  2. EXAMPLE The following table shows the cost per day for a rental car depending on how many miles you drive a day.

  3. LINEAR FUNCTIONS Definition: A linear function is a function of the form f (x) = mx + b. NOTE: The letters “m” and “b” represent coefficients (numbers).

  4. STRAIGHT LINES ANDLINEAR GRAPHS The graph of the linear function f (x) = mx + b is the straight line consisting of all the points (x,y) in the xy-plane that satisfy the equation y = mx + b.

  5. SLOPE AND y-INTERCEPTOFA LINE • The y-value where a line crosses the y-axis is called the y-intercept. • The slope of a line in the xy-plane is defined by

  6. SLOPE-INTERCEPTEQUATION • The constant b is the y-intercept of the line. • The coefficient m of x is the slope of the line. • This equation is called the slope-intercept equation for a line. In the equation y = mx + b,

  7. If (x0, y0) is a fixed point on a line, and (x, y) is any other point on the line, we can find the slope by The equation is called the point-slope equation for the line with slope m that passes through the point (x0, y0). POINT-SLOPE FORMULAFOR A LINE

  8. GRAPHS OF EQUATIONS Definition: The graph of an equation involving two variables x and y consists of all points in the xy-plane whose coordinates (x, y) satisfy the equation.

  9. GRAPHS OF LINES There are three types of equations whose graphs are straight lines: • Equations whose graphs are vertical lines. These have the form x = constant. • Equations whose graphs are horizontal lines. These have the form y = constant. • Equations whose graphs are slanted lines (lines that are neither vertical or horizontal). These have the form y = mx + b.

  10. GRAPH OF A FUNCTION Definition: The graph of the function f is the graph of the equation y = f (x).

  11. THE VERTICAL LINE TEST Recall that a function assigns to each number x a specific number f (x). Thus, a graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.

  12. EXAMPLE The following table shows the cost per day for a rental car depending on how many miles you drive a day. • How many miles did you drive if the cost was $80? • What do the slope and y-intercept mean in practical terms?

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