TABLE OF CONTENTS
Graph the following parabola.
We need a few points to graph this function.
The vertex and y- and x-intercepts are all relatively easy to find.
The vertex is at (3, 1).
The y-intercept is at x = 0. That is,
So, the y-intercept is (0, 7).
To find the x-intercepts, if they exist, we need to multiply everything out.
x-intercepts may not actually exist, just like Big Foot or Australia. Check the discriminate to see if they exist or not.
The discriminate is negative, so Australia isn't real. And neither are the x-intercepts for this function.
We do need a few more points, though. We're going to show x = 1 and x = 4, but whatever you choose should work out the same.
Collecting all of the points we've found into a table gives us the following.
The last three points come from the function's symmetry on the y-axis, centered on the vertex. For instance x = 6 is three away from the vertex, and so it shares a y value with (0, 7).
y = -3x2 + 1
Try to find the easy points first, to minimize how many calculations you have to do.
Notice that our function is equivalent to y = -3(x – 0)2 + 1. This means the vertex is at (0, 1).
We don't need to solve for the y-intercept, because it happens to be our vertex.
Looking at the discriminate, we see that 02 – 4(-3)(1) = 12. We have two real number roots, which means the function crosses the x-axis twice.
0 = -3x2 + 1
3x2 = 1
Our x-intercepts are and . This are equal to about (0.577, 0) and -0.577, 0).
We now have enough points to graph the function.
y = 2(x + 2)2 – 2
Start off by finding the vertex and the y-intercept.
The vertex is (-2, -2).
f(0) = 2(0 + 2)2 – 2 = 2(4) – 2 = 6
The y-intercept is (0, 6).
Expanding the function we find
0 = 2(x2 + 4x + 4) – 2
0 = x2 + 4x + 4 – 1
0 = x2 + 4x + 3
0 = (x + 3)(x + 1)
We would check the discriminate, but the function is so easily factored that we don't need to bother. That's pretty handy. The x-intercepts are (-3, 0) and (-1, 0).
And now, we graph.
We tell it to you straight.
It’s not all togas and Solo cups.