I have a graph here. The vertex is
(4,-4), then the points (3,-3) and
(5,-3), then the points on the x axis are (2,0) and (6,0), then the points (1,5) and (5,5). This is a parabola that opens up.
I'm asked to write a function for f(x) in standard form.
I'm a little stuck. Any help you can provide is appreciated. Thank you.
You should be familiar with the vertex form of a parabola as
y = a(x-p)^2 + q , where (p,q) is the vertex,
so we have:
y = a(x-4)^2 - 4
using any of the other given points will give us the a
let's use the easiest, (2,0)
0 = a(2-4)^2 - 4
0 = 4a - 4
a= 1
y = (x-4)^2 - 4 is your equation.
A quick mental check shows that all your other points satisfy this equation.
(x-4)^2 = k (y+4) from vertex
put in some other point like (2,0)
(2-4)^2 = k (4)
4 = 4 k
k = 1
so I propose (x-4)^2 = y + 4
check with another given point like (5,5)
Hey (5,5) should be (7,5) !!!Typo?
Try (3,-3) and (5,-3)
Thank you!
You are welcome :)
To write a function for f(x) in standard form for the given parabola, you can follow these steps:
Step 1: Determine the vertex of the parabola.
The vertex is given as (4, -4) in the form (h, k). In the standard form of a parabola, the vertex is expressed as (h, k).
Step 2: Identify the axis of symmetry.
The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is given by the equation x = h, where h is the x-coordinate of the vertex.
Step 3: Find the focus and directrix (optional).
These steps are not necessary to determine the equation in standard form, but they can provide additional information about the parabola. The focus is a point inside the parabola, and the directrix is a line outside the parabola. Their locations can be calculated using specific formulas.
Step 4: Determine whether the parabola opens upward or downward.
You mentioned that the parabola opens upward, so it will have an equation in the form f(x) = ax^2 + bx + c.
Step 5: Substitute the values of the vertex into the equation.
Using the vertex form, f(x) = a(x-h)^2 + k, substitute the x-coordinate (h) and y-coordinate (k) of the vertex into the equation.
With the given information about the parabola, let's proceed:
Step 1: The vertex is (4, -4).
Step 2: The axis of symmetry is x = 4.
Step 3: The focus and directrix can be calculated using specific formulas, but they are not needed to write the equation in standard form.
Step 4: The parabola opens upward, so the equation will be in the form f(x) = ax^2 + bx + c.
Step 5: Substituting the vertex coordinates into the vertex form, we have:
f(x) = a(x - 4)^2 - 4
To determine the value of a, we can use another point on the parabola. Let's use (3, -3).
Substituting these values into the equation:
-3 = a(3 - 4)^2 - 4
-3 = a(1)^2 - 4
-3 = a - 4
a = 1
Plugging the value of a back into the equation:
f(x) = 1(x - 4)^2 - 4
Simplifying the equation further:
f(x) = (x - 4)^2 - 4
Therefore, the equation for f(x) in standard form is f(x) = (x - 4)^2 - 4.