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Magazine Chapter 4: Problem 1
Given \(f(x)=2 x^{2}-x+6\), find \(f(-3), f(0)\), and \(f(10)\)
Short Answer
Expert verified
\(f(-3) = 27\), \(f(0) = 6\), \(f(10) = 196\)
Step by step solution
01
Identify the Function
The function given is \(f(x) = 2x^2 - x + 6\). This is a quadratic function, and we will substitute different values of \(x\) into this function to find \(f(-3)\), \(f(0)\), and \(f(10)\).
02
Evaluate f(-3)
Substitute \(x = -3\) into the function: \(f(-3) = 2(-3)^2 - (-3) + 6\). Calculate each term: \((-3)^2 = 9\) so \(2 \times 9 = 18\), \(-(-3) = 3\), and the constant is \(6\). Adding these gives \(f(-3) = 18 + 3 + 6 = 27\).
03
Evaluate f(0)
Substitute \(x = 0\) into the function: \(f(0) = 2(0)^2 - 0 + 6\). Simplify each term: \(2 \times 0 = 0\), \(0 = 0\), and the constant is \(6\). Thus, \(f(0) = 0 + 0 + 6 = 6\).
04
Evaluate f(10)
Substitute \(x = 10\) into the function: \(f(10) = 2(10)^2 - 10 + 6\). Calculate each term: \(10^2 = 100\) so \(2 \times 100 = 200\), \(-10 = -10\), and the constant is \(6\). Adding these gives \(f(10) = 200 - 10 + 6 = 196\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a handy tool to discover specific output values of a function given certain input values. It's like finding out which point on a curve corresponds to a particular x value. In simple terms, the substitution method involves replacing the variable in the function with a number or an expression. For instance, in our exercise, we substitute different values of \( x \) into the function \( f(x) = 2x^2 - x + 6 \).
Here's how it works in practice:
Here's how it works in practice:
- First, identify the given function. In this case, it is \( f(x) = 2x^2 - x + 6 \).
- Next, substitute the specific values for \( x \) into the function one by one.
- Perform the necessary arithmetic operations: calculate powers, multiply, add, or subtract numbers as needed.
- Finally, sum up all the calculated terms to get the final result.
Polynomial Functions
Polynomial functions are expressions that involve powers of a variable, like \( x \), with coefficients. The term "polynomial" comes from "poly," meaning many, and "nomials," meaning terms; so it literally means "many terms." The function \( f(x) = 2x^2 - x + 6 \) is a prime example of a quadratic polynomial function, mainly due to its highest exponent being \( 2 \).
Key features of polynomial functions include:
Understanding these characteristics helps in graphing polynomial functions and predicting their behavior over different ranges of \( x \). This forms the foundation for more advanced calculus concepts.
Key features of polynomial functions include:
- They have one or more terms, each consisting of a coefficient multiplied by a power of \( x \).
- The degree of the polynomial is determined by the highest power of \( x \); here, the degree is \( 2 \) as the highest power is \( x^2 \).
- The leading coefficient, which is the coefficient of the highest degree term, is \( 2 \) in this function.
- Polynomials are continuous and smooth curves, without any breaks or sharp turns.
Understanding these characteristics helps in graphing polynomial functions and predicting their behavior over different ranges of \( x \). This forms the foundation for more advanced calculus concepts.
Function Evaluation
Function evaluation refers to the process of calculating the output of a function for specific input values. It reveals what the function outputs at certain points on its graph. To perform a function evaluation, follow these steps:
In our exercise, we evaluated \( f(-3) \), \( f(0) \), and \( f(10) \), showing how changes in \( x \) affect \( f(x) \). By practice, function evaluation assists in visualizing how the function behaves, laying the groundwork for interpreting graphs and solving more complex mathematical problems.
- Identify the function you need to evaluate. In our example, the function \( f(x) = 2x^2 - x + 6 \) is provided.
- Substitute the given value of \( x \) into the function using the substitution method.
- Simplify the expression by performing arithmetic operations like multiplication, addition, and subtraction as necessary.
- The resulting value is the output of the function for that specific \( x \) value.
In our exercise, we evaluated \( f(-3) \), \( f(0) \), and \( f(10) \), showing how changes in \( x \) affect \( f(x) \). By practice, function evaluation assists in visualizing how the function behaves, laying the groundwork for interpreting graphs and solving more complex mathematical problems.
