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horizontal Intercepts and Related Topics

horizontal Intercepts and Related Topics - MAC1105...

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MAC1105: Horizontal Intercepts and Related Topics The points at which a graph intersects the horizontal axis are called the horizontal intercepts of the graph, and are the real number inputs that result in outputs of zero. These inputs are called zeros of the function. For example, the linear function f(x) 2x 10 = - has one zero at x 5 = , which means that f(5) 0 = and the point (5,0) is the x- intercept of the line y 2x 10 = - = . Algebraically we compute intercepts by letting the other variable in the formula equal 0 . When using the formula y 2x 10 = - , letting x equal 0 locates the y - intercept at -10 , and letting y equal 0 and solving the equation 2x 10 0 - = = locates the x -intercept at 5 . This last computation is relatively easy because the equation 2x 10 0 - = is linear. For other types of functions this can be a much more challenging process. We have been examining quadratic functions whose graphs are called parabolas . If the vertex is also the x - intercept, then the parabola will have only that point as a single x -intercept. But if the vertex is not on the x -axis, the parabola will have either two or no x -intercepts depending upon whether the arms of the parabola open toward or away from the x -axis. Finding these x -intercepts amounts to solving the quadratic equation 2 ax bx c 0 + + = + , and there are various methods for doing so. Algebraic methods include factoring and the quadratic formula . A graphing calculator can also be used, but will not find exact values if the intercepts occur at irrational numbers. Consider the quadratic equation 2 x 2x 15 0 - - = - = . This equation can be solved by factoring the quadratic and setting each factor equal to 0 . 2 x 2x 15 0 (x 3)(x 5) 0 x 3 0 x 5 0 x 3 x 5 - - = - - = - = - = + - = + - = - - + = - = = = - = = - = So the x -intercepts are at -3 and 5 . However, if the constant term is changed from -15 to -16 , the quadratic expression 2 x 2x 16 - - - - no longer factors. In this case we can use the quadratic formula 2 b b 4ac x 2a - ± - = . (See page 4 of this handout. Substituting into this formula (a 1, b 2, c 16) = = - = - results in the computation:
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( ) 2 ( 2) ( 2) 4(1)( 16) x 2(1) 2 4 64 2 68 2 4 17 2 2 2 2 1
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