Math Problem Statement

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

Solution

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(xh)2+ky = a(x - h)^2 + k

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

y=a(x3)2+4y = a(x - 3)^2 + 4

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

8=a(53)2+48 = a(5 - 3)^2 + 4

Simplifying:

8=a(2)2+48 = a(2)^2 + 4 8=4a+48 = 4a + 4 4=4a4 = 4a a=1a = 1

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

y=(x3)2+4y = (x - 3)^2 + 4

Sketch of the Graph:

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Graphing Parabolas

Formulas

Vertex form of a quadratic function: y = a(x - h)^2 + k

Theorems

Axis of symmetry of a parabola
Properties of quadratic functions

Suitable Grade Level

Grades 8-10