a Begin by writing an equation for the parabola. To do that you should write three equations to find the coefficients of parabola equation.
B
b Consider the equation that you wrote in Part A and use the formula for the axis of symmetry.
C
c What is coefficient of the quadratic term of the parabola equation? How does it affect the graph?
A
a a=22
b=26
c=6
d=2
B
b x=0
C
c Maximum
Practice makes perfect
a We will first find the equation for the parabola y=ax^2+bx+c. Then, we will use the equation to find the values of a, b, c, and d on the table. Be careful that the coefficients of the equation and the values on the table are not the same. There are for known points of the parabola.
(-20, -377), (-5,-2), (-1,22), (15,-202)
Because there are three coefficients, we should write three equations to find the coefficients. Let's use the last three points to write these equations. We will substitute them in the general form of the parabola equation.
Point
Substitution
Equation
( -5, -2)
-2=a( -5)^2+b( -5)+c
25a-5b+c=-2
( -1, 22)
22=a( -1)^2+b( -1)+c
a-b+c=22
( 15, -202)
-202=a( 15)^2+b( 15)+c
225a-15b+c=-202
Now we have three equations to write a system.
25a-5b+c=-2 & (I) a-b+c=22 & (II) 225a-15b+c=-202 & (III)Next we will solve the system. Look at Equation (II). We can begin by isolating a on the left-hand side and substitute it in the other two equations. Let's start because it will take some time.
Using the coefficients we found, we can complete the equation for the parabola.
y=- x^2+23
Now we are ready to find the unknowns on the table. We will first find the value of c by substituting (c,-13) in the equation.
The value of c is 6. Proceeding in the same way, we can find the other unknowns.
Point
y=- x^2+23
Unknown
( c, -13)
-13=- ( c)^2+23
c=6
( 5, a-24)
a-24=- ( 5)^2+23
a=22
( 7, - b)
- b=- ( 7)^2+23
b=26
We found a, b, and c. We didn't include d on the table because we have a special case for it. Let's determine the point that will help us to find d.
(d-1,a) ⇔ (d-1,22)
With this, we will substitute the point in the equation and find d.
Because the point (-1,22) is already on the table, the value of d can only be 2.
b To find x-coordinate of the vertex, we use the formula for the axis of symmetry, x=- b2a. Therefore, we will first highlight the coefficients of the equation in Part A.
y=- x^2+23 ⇔ y= -1 x^2+ 0x+ 23
By substituting the values in the formula, let's find the x-coordinate.
c In Part A, we found that the coefficient of the quadratic term is a=-1 which is negative. Therefore, the graph of the parabola opens downward and it has a maximum on its vertex.
