Question
Which is a quadratic function? \( \begin{array}{l}f(x)=2 x+x+3 \\ f(x)=0 x^{2}-4 x+7 \\ f(x)=5 x^{2}-4 x+5 \\ f(x)=3 x^{3}+2 x+2\end{array} \)
Ask by Gibbs Reeves. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The quadratic function is \( f(x) = 5x^{2} - 4x + 5 \).
Solution
A **quadratic function** is a polynomial of degree 2, generally in the form:
\[ f(x) = ax^{2} + bx + c \]
where \( a \neq 0 \).
Let's analyze each of the given functions:
1. **\( f(x) = 2x + x + 3 \)**
\[
f(x) = 3x + 3
\]
- **Degree:** 1 (Linear function)
2. **\( f(x) = 0x^{2} - 4x + 7 \)**
\[
f(x) = -4x + 7
\]
- **Degree:** 1 (Linear function)
3. **\( f(x) = 5x^{2} - 4x + 5 \)**
- **Degree:** 2 (Quadratic function)
4. **\( f(x) = 3x^{3} + 2x + 2 \)**
- **Degree:** 3 (Cubic function)
**Conclusion:**
The quadratic function among the listed options is:
\[
f(x) = 5x^{2} - 4x + 5
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
A quadratic function is a polynomial function of degree 2, which means its highest power of \( x \) is 2. In the options provided, the function \( f(x) = 5x^2 - 4x + 5 \) is the quadratic function because it contains the \( x^2 \) term, while the others either lack this or have higher-degree terms. Quadratic functions can often resemble a U-shaped graph called a parabola. The vertex of the parabola gives you crucial information about the maximum or minimum point of the function. Plus, when you have a positive leading coefficient as in this case, the parabola opens upwards, making it a happy little curve!
