Write the equation of the parabola in vertex form given the points below.

Question

[MAKE SURE to first solve for the a, minus, v, a, l, u, ea−value. Also, you must type f, of, x, equalsf(x)= for your equation.]

Vertex left parenthesis, 4, comma, 6, right parenthesis(4,6) and the x-intercept is at left parenthesis, 1, comma, 0, right parenthesis(1,0)

Answer 1

To find the equation of the parabola in vertex form, we start with the general vertex form of a parabola, which is:

\[ f(x) = a(x - h)^2 + k \]

where \((h, k)\) is the vertex of the parabola. Given the vertex \((4, 6)\), we have \(h = 4\) and \(k = 6\). Thus, we can write:

\[ f(x) = a(x - 4)^2 + 6 \]

Next, we need to determine the value of \(a\). To do this, we'll use the x-intercept provided, which is \((1, 0)\). This means that when \(x = 1\), \(f(x) = 0\). We can substitute these values into the equation to solve for \(a\):

\[ 0 = a(1 - 4)^2 + 6 \]

Calculating \(1 - 4\):

\[ 1 - 4 = -3 \]

Now substituting:

\[ 0 = a(-3)^2 + 6 \]

Simplifying \((-3)^2 = 9\):

\[ 0 = 9a + 6 \]

Next, we isolate \(a\):

\[ 9a = -6 \]

Dividing both sides by 9 gives:

\[ a = -\frac{2}{3} \]

Now, substituting \(a\) back into our vertex form equation:

\[ f(x) = -\frac{2}{3}(x - 4)^2 + 6 \]

Thus, the equation of the parabola in vertex form is:

\[ f(x) = -\frac{2}{3}(x - 4)^2 + 6 \]