Math Problem Statement

The quadratic function g(x) = -x^2 + 4x - 7 can be rewritten in the form y = a(x - h)^2 + k. Find the real numbers a, h, and k. Also give the two coordinates of the vertex.

Solution

We are given the quadratic function:

g(x)=x2+4x7g(x) = -x^2 + 4x - 7

This can be rewritten in the form:

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

To do this, we will complete the square to transform the equation into the vertex form.

Step 1: Factor out the coefficient of x2x^2

The equation is g(x)=x2+4x7g(x) = -x^2 + 4x - 7. First, we factor out 1-1 from the terms involving xx:

g(x)=(x24x)7g(x) = - (x^2 - 4x) - 7

Step 2: Complete the square

To complete the square inside the parentheses, take half of the coefficient of xx, square it, and add it inside the parentheses, while compensating by subtracting the same value outside the parentheses. Half of 4-4 is 2-2, and (2)2=4(-2)^2 = 4.

We add and subtract 4 inside the parentheses:

g(x)=(x24x+44)7g(x) = - (x^2 - 4x + 4 - 4) - 7

Simplify:

g(x)=[(x24x+4)4]7g(x) = - [(x^2 - 4x + 4) - 4] - 7

g(x)=(x2)2+47g(x) = - (x - 2)^2 + 4 - 7

Step 3: Simplify

Now, simplify the constants:

g(x)=(x2)23g(x) = - (x - 2)^2 - 3

Step 4: Identify aa, hh, and kk

From the vertex form y=a(xh)2+ky = a(x - h)^2 + k, we can see that:

  • a=1a = -1
  • h=2h = 2
  • k=3k = -3

Step 5: Determine the vertex

The vertex of the quadratic function occurs at (h,k)(h, k). Therefore, the vertex is at:

(2,3)(2, -3)

Final Answer:

  • a=1a = -1
  • h=2h = 2
  • k=3k = -3
  • The vertex is at (2,3)(2, -3)

Let me know if you would like more details, or if you have any questions.


Here are 5 related questions for you to consider:

  1. How do you complete the square for a general quadratic function?
  2. What is the significance of the vertex in a quadratic function?
  3. How does the value of aa affect the direction of the parabola's opening?
  4. Can a quadratic function have more than one vertex?
  5. How can the vertex form of a quadratic be used to determine the axis of symmetry?

Tip: The vertex form of a quadratic is particularly useful when you need to quickly identify the vertex and the axis of symmetry!

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Form
Completing the Square

Formulas

Standard form of a quadratic: ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k

Theorems

Quadratic Function Transformation
Completing the Square

Suitable Grade Level

Grades 9-12