Sketch the graph of the quadratic function with the given vertex and that passes through the given point. Vertex (3,4) ; Point (5,8)

To find the equation of the quadratic function and sketch the graph, we can use the **vertex form** of a quadratic equation, which is:

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

where \((h, k)\) is the vertex. For this problem, the vertex is given as \((3, 4)\), so the equation becomes:

\[
y = a(x - 3)^2 + 4
\]

We are also given that the function passes through the point \((5, 8)\). To find the value of \(a\), substitute \(x = 5\) and \(y = 8\) into the equation:

\[
8 = a(5 - 3)^2 + 4
\]

Simplifying:

\[
8 = a(2)^2 + 4
\]
\[
8 = 4a + 4
\]
\[
4 = 4a
\]
\[
a = 1
\]

Now that we have \(a = 1\), the equation of the quadratic function is:

\[
y = (x - 3)^2 + 4
\]

### Sketch of the Graph:

1. The vertex is at \((3, 4)\).
2. The parabola opens **upwards** because \(a = 1\), which is positive.
3. It passes through the point \((5, 8)\).
4. The axis of symmetry is the vertical line \(x = 3\).

Would you like details on how to sketch it step by step or explore further?

Here are 5 related questions for practice:
1. What is the axis of symmetry of a quadratic function?
2. How does changing the value of \(a\) affect the graph of a quadratic function?
3. How do you find the x-intercepts of a quadratic function?
4. Can you determine the direction of a parabola based on the sign of \(a\)?
5. How can you convert a quadratic function from vertex form to standard form?

**Tip**: The vertex form of a quadratic function is especially useful when you know the vertex, as it simplifies the process of graphing and solving equations.

Learn how to sketch the graph of a quadratic function with vertex (3, 4) and passing through point (5, 8). This problem covers key algebraic concepts like vertex form and graphing parabolas. Suitable for students in Grades 8-10.

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