Math Problem Statement

Find the vertex of f(x) = x^2 - 10x + 28. After this is correctly completed, you will be asked to make a table for the function and graph it. The vertex has been found as (5,3). Complete the table by finding the values of f(x) at x = 4 and x = 6.

Solution

The given quadratic function is f(x)=x2−10x+28f(x) = x^2 - 10x + 28. The vertex of this parabola has already been identified as (5,3)(5, 3), and you are tasked with filling in the table for the values of f(x)f(x) at x=4x = 4 and x=6x = 6.

Let's find f(4)f(4) and f(6)f(6).

For x=4x = 4:

f(4)=42−10(4)+28=16−40+28=4f(4) = 4^2 - 10(4) + 28 = 16 - 40 + 28 = 4

For x=6x = 6:

f(6)=62−10(6)+28=36−60+28=4f(6) = 6^2 - 10(6) + 28 = 36 - 60 + 28 = 4

Now, you can complete the table as follows:

\hline x & f(x) \\ \hline 4 & 4 \\ 5 & 3 \\ 6 & 4 \\ \hline \end{array}$$ If you have any questions or need further clarification, feel free to ask! Here are five related questions to expand your understanding: 1. How can you find the axis of symmetry for a quadratic function? 2. What is the significance of the vertex in the context of graphing a quadratic function? 3. How do you determine if the vertex represents a minimum or maximum point? 4. How would the graph change if the coefficient of $$x^2$$ were negative? 5. Can you use a different method (like factoring or completing the square) to solve for the vertex? **Tip**: The vertex form of a quadratic equation, $$f(x) = a(x-h)^2 + k$$, makes it easy to identify the vertex directly.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Vertex Form
Graphing

Formulas

Vertex formula for a quadratic: x = -b / 2a
Standard form of a quadratic equation: ax^2 + bx + c

Theorems

Vertex Theorem for Parabolas

Suitable Grade Level

Grades 9-10

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