Math Problem Statement

The function f(x) = x^2 - 6x + 5 represents a parabola. Rewrite the function in vertex form and find the vertex coordinates.

Solution

The problem asks you to rewrite the given function f(x)=x26x+5f(x) = x^2 - 6x + 5 in vertex form and use it to determine the vertex of the parabola. Let's solve this step-by-step:

1. Rewrite f(x)=x26x+5f(x) = x^2 - 6x + 5 in vertex form using completing the square.

The vertex form of a quadratic function is: f(x)=a(xh)2+kf(x) = a(x-h)^2 + k where (h,k)(h, k) is the vertex.

Step 1. Factor out a=1a = 1 (if necessary, but here a=1a = 1 already):

f(x)=x26x+5f(x) = x^2 - 6x + 5

Step 2. Complete the square:

To complete the square, take half of the coefficient of xx (which is 6-6), square it, and add/subtract it inside the equation: Half of 6=3,(3)2=9\text{Half of } -6 = -3, \quad (-3)^2 = 9

Add and subtract 9 inside the equation: f(x)=(x26x+9)9+5f(x) = (x^2 - 6x + 9) - 9 + 5

Simplify: f(x)=(x3)24f(x) = (x-3)^2 - 4

Final vertex form:

f(x)=(x3)24f(x) = (x-3)^2 - 4

2. Determine the vertex:

From the vertex form f(x)=(x3)24f(x) = (x-3)^2 - 4, the vertex is: (h,k)=(3,4)(h, k) = (3, -4)


3. Match this to the answer choices:

  • The correct function in vertex form is f(x)=(x3)24f(x) = (x-3)^2 - 4.
  • The vertex coordinates are (3,4)(3, -4).

4. Final Answer:

  • Select f(x)=(x3)24f(x) = (x-3)^2 - 4.
  • The vertex is (3,4)(3, -4).

Would you like further clarification or additional examples? Let me know!


Relative Questions:

  1. What does it mean for a quadratic function to be in vertex form?
  2. How do you complete the square for any quadratic equation?
  3. Why is the vertex useful in understanding the graph of a parabola?
  4. What is the significance of the hh and kk in the vertex form equation?
  5. How do you determine whether a parabola opens upward or downward?

Tip:

The coefficient of (xh)2(x-h)^2 in the vertex form (aa) determines the "width" and direction of the parabola. If a>0a > 0, the parabola opens upwards, and if a<0a < 0, it opens downwards!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Completing the Square
Vertex Form of a Parabola

Formulas

Vertex form: f(x) = a(x - h)^2 + k
Completing the square: Add and subtract (b/2)^2 inside the equation

Theorems

Vertex Formula: The vertex of f(x) = a(x - h)^2 + k is (h, k).

Suitable Grade Level

Grades 8-10