Create a graph that intersects f(x) in its vertex but opens up in the opposite direction.
B
b
Can we manipulate the function from Part A to make a new function that does not intersect f(x)?
A
a
Example Solution: g(x)=- 3(x+4)^2-8
B
b
Example Solution: h(x)=- 3(x+4)^2-9
Practice makes perfect
a
A function in graphing form is written in the following format.
y=a(x- h)^2+ k
In this form, the function's vertex can be identified as ( h, k). Let's rewrite f(x) so that it matches this form exactly.
f(x)=3(x-( - 4))^2+( -8)
Having rewritten the equation in graphing form, we see that f(x) has a vertex in ( -4, - 8). Also, since a=3 we know that x^2 must have a positive coefficient, which means the parabola of f(x) opens upward. With this information, we can sketch the function's graph.
To draw a function g(x) that intersect f(x) in only one point, we can, for example, change the sign of a which produces a parabola with a maximum value at (-4,- 8). In other words, the parabola of g(x) will open downwards.
As we can see, the graphs intersect at only one point.
b
In Part A, we created a function that intersected f(x) once in its vertex. To create a second equation h(x) that does not intersect f(x) at all, we can vertically translate g(x) in the negative direction by, for example, 1 unit.
h(x)= - 3(x+4)^2-8 - 1
↓
h(x)=- 3(x+4)^2-9
The graph of h(x) has a vertex in (-4,-9) and opens up downward. Therefore, it will never intersect f(x). Let's show the graph.
Note that both solutions are only examples, and there are infinitely many other possible solutions.
