Graphing quadratic functions

Question

1.)1. Explain the procedure (but don't create the graph) for graphing the quadratic function in vertex form f(x) = 2(x - 3)² - 1 using pencil and paper. Your explanation should show your understanding of using the perfect squares pattern from the vertex. Explain the entire process.

2. Explain the procedure(but don't create the graph) for graphing the quadratic function in factored form f(x) = (x - 2)(x + 4) using pencil and paper. Your explanation should show your understanding of the intercept graphing method. Explain the entire process.

3. Explain how a function graph can be used to help verify that a function is factored correctly.

Answer 1

1. Since the equation uses x² you know this has to be a parabola. The vertex of the parabola can be found by using the two values given to you in the equation given. To find the x value of the vertex, you would find at what value would whatever you are squaring be 0, so x-3=0. This value is 3, so the x value is 3. To find the y value, you would use the c value, in this case -1. So, the vertex is (3,-1). To find the points where it crosses the x axis, you would find the x value where y=0. So, 0=2(x-3)²-1. From solving this formula you would get the values of 2.293,0 and 3.707,0. This is how you would graph the parabola.
2. For this problem, you can immediately tell from the equation where the two x intercepts are. You are trying to find at what values f(x)=0, so you set x-2=0 and x+4=0. So the x intercepts are (2,0) and (-4,0). After this, you can find the vertex by foiling the equation out, so you end up with x²+2x-8. Vertex of a parabola is equal to x= -b/2a. So in this case the x value is -2/2=-1. So the x value is -1. When you plug this in to the equation you find that at x=-1, y=1-2-8=-9. So the vertex is (-1,-9)
3. Using a function graph, you can find your x intercept values by looking at where the graph crosses the x axis. If you factored an equation correctly, the x intercept values should be equal to the value if you set your factors equal to 0. So if one of your factors is x-3, then your x intercept value should be 3 because if you set x-3=0, your x value is 3. If this is the case, you know you factored correctly.