Question

which quadratic function in vertex form can be represented by the graph that has a vertex at (1, 46) and passes through the point (3, 10)?

Answer 1

To find the quadratic function in vertex form that has a vertex at \( (1, 46) \) and passes through the point \( (3, 10) \), we can start using the vertex form of a quadratic equation, which is given by:

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

where \( (h, k) \) is the vertex of the parabola. In this case, the vertex is \( (1, 46) \), so we can substitute \( h \) and \( k \) into the equation:

\[
f(x) = a(x - 1)^2 + 46
\]

Next, we need to determine the value of \( a \) using the point \( (3, 10) \) that lies on the graph. We can substitute \( x = 3 \) and \( f(x) = 10 \) into the equation:

\[
10 = a(3 - 1)^2 + 46
\]

This simplifies to:

\[
10 = a(2)^2 + 46
\]
\[
10 = 4a + 46
\]

Now, we isolate \( a \):

\[
10 - 46 = 4a
\]
\[
-36 = 4a
\]
\[
a = -9
\]

Now that we have the value of \( a \), we can write the final equation of the quadratic function:

\[
f(x) = -9(x - 1)^2 + 46
\]

This is the quadratic function in vertex form that meets the given conditions.