Math Problem Statement

Given the quadratic equation f(x) = x^2 - 4x + 2, fill in the table, calculate values for f(x) at specific x, and identify the correct graph of the equation.

Solution

The given image contains a quadratic function problem where f(x)=x2−4x+2f(x) = x^2 - 4x + 2, and you are tasked with filling a table of values and selecting the correct graph corresponding to the equation. I'll assist you step-by-step:

Step 1: Filling the Table

The given function is: f(x)=x2−4x+2f(x) = x^2 - 4x + 2

Let's compute f(x)f(x) for x=1,2,3,4x = 1, 2, 3, 4.

  • For x=1x = 1: f(1)=12−4(1)+2=1−4+2=−1f(1) = 1^2 - 4(1) + 2 = 1 - 4 + 2 = -1

  • For x=2x = 2: f(2)=22−4(2)+2=4−8+2=−2f(2) = 2^2 - 4(2) + 2 = 4 - 8 + 2 = -2

  • For x=3x = 3: f(3)=32−4(3)+2=9−12+2=−1f(3) = 3^2 - 4(3) + 2 = 9 - 12 + 2 = -1

  • For x=4x = 4: f(4)=42−4(4)+2=16−16+2=2f(4) = 4^2 - 4(4) + 2 = 16 - 16 + 2 = 2

So, the corrected table is:

\hline x & f(x) & (x, y) \\ \hline 1 & -1 & (1, -1) \\ 2 & -2 & (2, -2) \\ 3 & -1 & (3, -1) \\ 4 & 2 & (4, 2) \\ \hline \end{array}$$ --- ### Step 2: Graph Matching The function $$f(x) = x^2 - 4x + 2$$ is a **parabola that opens upwards** because the coefficient of $$x^2$$ is positive. Observing the vertex and symmetry: - The vertex occurs at $$x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$$. - At $$x = 2$$, the function reaches its minimum value ($$f(2) = -2$$). - The graph will be symmetric around $$x = 2$$. Thus, the correct graph from the choices is **Graph C**. --- ### Final Answer: 1. The completed table: $$\begin{array}{|c|c|c|} \hline x & f(x) & (x, y) \\ \hline 1 & -1 & (1, -1) \\ 2 & -2 & (2, -2) \\ 3 & -1 & (3, -1) \\ 4 & 2 & (4, 2) \\ \hline \end{array}$$ 2. The matching graph is **Graph C**. --- Let me know if you want a detailed explanation of the graph selection or have any other questions! 😊 --- ### Related Questions: 1. What happens to the shape of the graph if the $$x^2$$ coefficient is negative? 2. How would the vertex change if we alter the linear term $$b$$ in $$ax^2 + bx + c$$? 3. How do you find the axis of symmetry in a quadratic equation? 4. Can quadratic functions ever have no real roots? How can you tell from the graph? 5. What is the effect of changing $$c$$ in $$ax^2 + bx + c$$ on the graph? ### Tip: When graphing quadratics, always calculate the vertex using $$-\frac{b}{2a}$$ to understand the graph's symmetry and center point!

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Graphing Parabolas
Vertex Form
Axis of Symmetry

Formulas

f(x) = ax^2 + bx + c
Vertex: x = -b / (2a)
Symmetry in quadratic graphs

Theorems

Properties of Quadratic Functions
Axis of Symmetry Theorem

Suitable Grade Level

Grades 8-10

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