A. A parabola is the graph of a quadratic equation. In other words, it is the graph of the function y = ax2 + bx + c, sometimes written y = a(x - h)2 + k.
B. When dealing with parabolas we are concerned with 3 things: x-intercepts, y-intercept, and the vertex.
C. X-intercepts
1. To find out how many x-intercepts there are, use the discriminant: b2 - 4ac. If the discriminant = 0, then there is 1 x-intercept. If the discriminant > 0, then there are 2 x-intercepts. If the discriminant < 0, there are 0 x-intercepts.
2. To find the x-intercepts, factor the equation and solve for y = 0. So the answer will always be (n,0) where n is the value(s) you get when solving for 0.
Ex [1] The graph of the equation y = 2x2 - 4x + 3 has how many x-intercepts? _____
a. We use the discriminant: b2 - 4ac.
b. (-4)2 - 4(2)(3) = 16 - 24 = -8. Since this value is less than 0, the answer is 0 x-intercepts.
Ex [2] If the smallest x-intercept of the function y = x2 - 4x - 5 is (a,b) then a = ______
a. This time, we should factor the equation and solve for y = 0.
b. Factoring we get: (x - 5) (x + 1) = 0. Solving both expressions we get x = 5 and x = -1. The smallest value is -1.
c. The x-intercept is (-1,0) so a = -1.
*Note: If a question ever asks for the 'b' value, the answer is always 0.
D. Y-intercepts
1. Y-intercepts are easy. Simply use x=0 and solve for y. If the expression is written as ax2 + bx + c, the answer is (0,c). Otherwise, the answer is (0,n), where n is the value you get when solving for x = 0.
Ex [1] The y-intercept of the function y = 5x2 - 3x + 4 is (a,b). Then b= _____
a. The answer is simply 4.
Ex [2] The y-intercept of the function y = 2(x-3)2 + 2 is (a,b). Then b= _____
a. In this case, use x = 0 and solve for y.
b. 2(0-3)2 + 2 = 2(9) + 2 = 20.
c. The answer is 20.
*Note: If the question ever asks for the 'a' value, the answer is always 0.
E. Vertex
1. The vertex of the parabola is the highest (or lowest) point in the graph. It is written as the point (h,k). If the equation is written as y = a(x - h)2 + k, finding the vertex is very easy. However, if the equation is written as y = ax2 + bx + c, then do the following:
a. To find 'h', use: -b/2a.
b.
To find 'k', you can plug 'h' into the equation and solve for y, or you can use:
. Notice,
the numerator is the same formula as the discriminant so this equation should be
easy to remember.
Ex [1] If the vertex of the parabola, y = 2(x-3)2 + 4 is (h,k), then k = _____.
a. The answer is simply 4.
Ex [2] If the vertex of the parabola, y = 4x2 - 5x - 1 is (h,k), then k = _____.
a. To find k, we need to first find the discriminant or (-5)2 - 4(4)(-1) = 25 + 16 = 41.
b. The denominator is 4a = 4(4) = 16.
c. The answer is -41/16.
d. If the question had asked for 'h' the answer would be -b/2a or -(-5)/2(4) = 5/8.