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Chapter 1 | Functions and Graphs
Figure 1.19 (a) For a quadratic function, if the leading coefficient a >0, the parabola opens upward. If a <0, the parabola opens downward. (b) For a cubic function f , if the leading coefficient a >0, the values f ( x )→∞ as x →∞ and the values f ( x )→−∞ as x →−∞. If the leading coefficient a <0, the opposite is true.
Zeros of Polynomial Functions Another characteristic of the graph of a polynomial function is where it intersects the x -axis. To determine where a function f intersects the x -axis, we need to solve the equation f ( x ) =0 for x . In the case of the linear function f ( x ) = mx + b , the x -intercept is given by solving the equation mx + b =0. In this case, we see that the x -intercept is given by (− b / m , 0). In the case of a quadratic function, finding the x -intercept(s) requires finding the zeros of a quadratic equation: ax 2 + bx + c =0. In some cases, it is easy to factor the polynomial ax 2 + bx + c to find the zeros. If not, we make use of the quadratic formula.
Rule: The Quadratic Formula Consider the quadratic equation
ax 2 + bx + c =0, where a ≠0. The solutions of this equation are given by the quadratic formula
(1.8)
2 −4 ac
x = − b ± b
2 a . If the discriminant b 2 −4 ac >0, this formula tells us there are two real numbers that satisfy the quadratic equation. If b 2 −4 ac =0, this formula tells us there is only one solution, and it is a real number. If b 2 −4 ac <0, no real numbers satisfy the quadratic equation. In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the x -axis. In some instances, it is possible to find the x -intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the x -intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the x -intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly.
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