a The equation resembles a function written in graphing form.
B
b At the y-intercept we have x=0. At the x-intercept we have y=0.
A
a See solution.
B
b x-intercepts: None
y-intercepts: (0,20)
Practice makes perfect
a We are asked to help find information and make a sketch of our friend's function. Examining it, we notice that it resembles a function written in graphing form. This makes it easy to find its vertex.
Graphing Form:& y=a(x-h)^2+k
Vertex:& (h,k)
However, our function does not resemble this form exactly. Therefore, we must first make sure that it does so we can determine its vertex.
Function:& y=2(x-3)^2-2
Vertex:& (3,2)
Notice that the value of a tells us the function's vertical stretch and also whether it opens up or down. Since the value of a is 2 — a positive number — this parabola opens upward and is vertically stretched by a factor of 2. With this information, we can make a sketch of the parabola.
b We now want to find the x- and y-intercepts without drawing an accurate graph or using a calculator. We know that a function intercepts the y-axis when x=0. Therefore, let's substitute x with 0 and evaluate the right-hand side.
Notice that we have a negative number on the left-hand side of the equation and a square of some number on the right-hand side. The square of any real number is non-negative, so it is impossible for these two expressions to be equal to each other if x is a real number. For this reason there are no x-intercepts.
