Use the vertex form. How do the parameters of the vertex form affect the shape of the parent function, y=x^2?
Example equations: f_1(x) = 12(x-3)^2 and f_2(x) = - 12(x-3)^2
Practice makes perfect
The parent function for a quadratic function is y=x^2. Its graph is a parabola.
We can write any quadratic equation in vertex form. The vertex form is a transformation of the parent function.
f(x) = a(x-h)^2 +k
In this form, the vertex is located at (h,k), and a is a parameter that can stretch or compress the parabola vertically or perform a reflection across the x-axis. Then, we can ensure that the vertices coincide if we use the same values for k and h in both equations. For instance, h=3 and k= .
f_1(x) = a_1(x)(x-3)^2 +
f_2(x) = a_2(x)(x-3)^2 +
With this our parabolas will lay on the x-axis. Now, recall that the magnitude of a tells us if the parent function will be stretched or compressed in the y-axis. Therefore, the magnitude of a just affects the shape of the parabola.
If |a| > 1 &the graph gets vertically
&stretched, looking thinner.
If |a| < 1 &the graph gets vertically
&compressed, looking wider.
Nevertheless, the sign a determines if the parabola opens up or downwards. Therefore, to have two parabolas such that one is the reflection of the other across the x-axis, their a parameter has to be of the same magnitude, but opposite signs. For instance, a_1=12 and a_2= - 12.
f_1(x) = 12(x-3)^2
f_2(x) = - 12(x-3)^2
Notice that we could have used different values for the parameters of the vertex form equation. There are infinitely many possible solutions fulfilling this exercise's requirements, therefore, this is just an example solution.
