Math Problem Statement

Determine the quadratic function whose graph is given with vertex (-3, -4) and point (0, 5).

Solution

To find the quadratic function that represents the given graph, let’s analyze the provided information.

Step-by-Step Solution:

  1. Vertex Form of a Quadratic Function: The vertex form of a quadratic function is given by:

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

    where:

    • (h,k)(h, k) is the vertex of the parabola.
    • aa determines the shape and direction (upward or downward) of the parabola.
  2. Identify the Vertex: From the graph, the vertex is clearly labeled as (3,4)(-3, -4). So, h=3h = -3 and k=4k = -4.

  3. Substitute the Vertex into the Equation: The equation in vertex form becomes:

    f(x)=a(x+3)24f(x) = a(x + 3)^2 - 4

  4. Find the Value of aa: To find aa, use another point on the graph. From the graph, a clear point is (0,5)(0, 5).

    Plugging in x=0x = 0 and f(0)=5f(0) = 5:

    5=a(0+3)245 = a(0 + 3)^2 - 4 Simplifying:

    5=a(9)45 = a(9) - 4 5+4=9a5 + 4 = 9a 9=9a9 = 9a a=1a = 1

  5. Final Equation: The equation of the quadratic function is:

    f(x)=(x+3)24f(x) = (x + 3)^2 - 4

This is the quadratic function that represents the given graph.

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

  1. How would the graph change if aa were negative?
  2. What is the significance of the vertex in a quadratic function?
  3. How do we convert from the vertex form to the standard form of a quadratic equation?
  4. What other points on the graph can we use to verify a=1a = 1?
  5. How does changing hh or kk affect the position of the parabola?

Tip: Always use a clear point on the graph, preferably an intercept, to solve for aa accurately.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions

Formulas

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

Theorems

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Suitable Grade Level

Grades 9-12