Question

1. the graph shows the height of the flying disk with respect to time. what is the equation of the function? write the equation in vertex form. then write the equation in the form y = ax² + bx + c.

Explanation:

Step1: Recall vertex - form of a quadratic function

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. First, we need to identify the vertex from the graph.

Step2: Determine the value of \(a\)

We can use another point on the graph, substitute its \(x\) and \(y\) values into the vertex - form equation \(y=a(x - h)^2+k\) along with the known \(h\) and \(k\) values to solve for \(a\).

Step3: Expand the vertex - form to standard form

To convert from vertex - form \(y=a(x - h)^2+k\) to standard form \(y = ax^{2}+bx + c\), we expand \((x - h)^2=x^{2}-2hx+h^{2}\) and then distribute \(a\) and combine like terms.

However, since the graph is not provided, we cannot perform the actual calculations. But the general process to find the equations is as described above.

If we assume the vertex \((h,k)\) is known and a point \((x_1,y_1)\) on the graph is known:

1. Substitute \((h,k)\) into \(y=a(x - h)^2+k\) to get \(y=a(x - h)^2+k\).
2. Substitute \((x_1,y_1)\) into \(y=a(x - h)^2+k\): \(y_1=a(x_1 - h)^2+k\). Then solve for \(a\):

\[a=\frac{y_1 - k}{(x_1 - h)^2}\]

1. For standard form, start with \(y=a(x - h)^2+k=a(x^{2}-2hx + h^{2})+k=ax^{2}-2ahx+ah^{2}+k\). So \(b=-2ah\) and \(c = ah^{2}+k\)

Since we don't have the graph details, we can't give a specific answer. But if we had a vertex \((h,k)=(2,3)\) and a point \((0,1)\) on the parabola:

Step1: Write the vertex - form

The vertex - form is \(y=a(x - 2)^2+3\)

Step2: Find \(a\)

Substitute \(x = 0\) and \(y = 1\) into \(y=a(x - 2)^2+3\):
\[1=a(0 - 2)^2+3\]
\[1 = 4a+3\]
\[4a=-2\]
\[a=-\frac{1}{2}\]
The vertex - form is \(y=-\frac{1}{2}(x - 2)^2+3\)

Step3: Convert to standard form

\[y=-\frac{1}{2}(x^{2}-4x + 4)+3\]
\[y=-\frac{1}{2}x^{2}+2x-2 + 3\]
\[y=-\frac{1}{2}x^{2}+2x + 1\]

Answer:

Without the graph, we cannot provide a specific equation. But the general vertex - form is \(y=a(x - h)^2+k\) and standard - form is \(y = ax^{2}+bx + c\) where the values of \(a\), \(b\), and \(c\) are found as described above.