242 CHAPTER 2 Graphs and Functions (c) For 15, -32 and 1-2, -32, the slope equals 0. m= -3 - 1-32 -2 - 5 = 0 -7 = 0 A sketch would show that the line through 15, -32 and 1-2, -32 is horizontal. S Now Try Exercises 41, 47, and 49. LOOKING AHEAD TO CALCULUS The derivative of a function provides a formula for determining the slope of a line tangent to a curve. If the slope is positive on a given interval, then the function is increasing there. If it is negative, then the function is decreasing. If it is 0, then the function is constant. Slopes of Horizontal and Vertical Lines The slope of a horizontal line is 0. The slope of a vertical line is undefined. Theorems for similar triangles can be used to show that the slope of a line is independent of the choice of points on the line. That is, slope is the same no matter which pair of distinct points on the line are used to find it. If the equation of a line is in the form y = ax + b, we can show that the slope of the line is a, the coefficient of x. To do this, we use function notation and the definition of slope. m= ƒ1x22 - ƒ1x12 x2 - x1 Slope formula m= 3a1x + 12 + b4 - 1ax + b2 1x + 12 - x Let ƒ1x2 = ax + b, x1 = x, and x2 = x + 1. m= ax + a + b - ax - b x + 1 - x Distributive property m= a 1 Combine like terms. m= a The slope is a. This discussion enables us to find the slope of the graph of any linear equation by solving for y and identifying the coefficient of x, which is the slope. EXAMPLE 6 Finding Slope from an Equation Find the slope of the line 4x + 3y = 12. SOLUTION Solve the equation for y. 4x + 3y = 12 3y = -4x + 12 Subtract 4x. y = - 4 3 x + 4 Divide by 3. The slope is - 4 3 , which is the coefficient of x when the equation is solved for y. S Now Try Exercise 55(a). Be careful to divide each term by 3. The results in Examples 5(b) and 5(c) suggest the following generalizations.
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