Exercise 69 Page 394

Let's start by finding the y-intercept. This is given by the function's constant. y=x^2+x - 6 The graph intercepts the y-axis at (0,- 6). To find the x-intercepts we should substitute y with 0 and then solve for x.

To solve this equation, we have to write the equation in factor form. y=(x+a)(x+b) If we can write the function in factor form we can then use the Zero Product Property to find the x-intercepts. If this particular function can be factored we should be able to find two terms, a and b, where ab=-6 and (a+b)=1 y=x^2+ 1x - 6 y=x^2+ (a+b)x+ ab Let's factor - 6 in as many ways as we can and add the factors. When the sum equals 1 we have identified the correct factors, a and b. c|c|c|c|c Product & a(b) & a+b & Sum & Equals1? [0.5em] - 6 & - 6(1) & - 6+1& - 5 & * [0.1em] - 6 & - 1(6) & - 1+6& 5 & * [0.1em] - 6 & - 3(2) & - 3+2& - 1 & * [0.1em] - 6 & - 2(3) & - 2+3& 1 & ✓ When a= - 2 and b= 3, the sum of a and b equals 1. Let's substitute these values into the factor form of the equation. y=(x+( - 2))(x+ 3) ⇓ y=(x-2)(x+3) The Zero Product Property can now be used by setting each of our factors equal to zero and solving for x. x-2=0 &⇔ x=2 x+3=0 &⇔ x=-3 We have enough information to sketch the graph.