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Magazine Chapter 9: Problem 17
If \(f(x)=5 x^{2}-3 x-27,\) then \(f(-4)=?\) A. 385 B. 172 C. 65 D. 41 E. -95
Short Answer
Expert verified
C. 65
Step by step solution
01
Write down the function
The given function is \(f(x) = 5x^2 - 3x - 27\).
02
Substitute x with -4
To find \(f(-4)\), replace \(x\) with \(-4\) in the function:
\(f(-4) = 5(-4)^2 - 3(-4) - 27\)
03
Simplify the expression
Now, calculate each term:
\((-4)^2 = 16\),
\(5 \cdot 16 = 80\),
\(-3 \cdot (-4) = 12\),
So, the expression becomes:
\(f(-4) = 80 + 12 - 27 \)
Now, add and subtract the values:
\(f(-4) = 80 + 12 - 27= 92 - 27\)
\(f(-4) = 65\)
04
Compare the calculated value to the options
The calculated value of \(f(-4)\) is \(65\). Comparing this value with the given options, the correct answer is:
C. 65
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Evaluation
Function evaluation is a fundamental concept that involves replacing the variable in a function expression with a specific value. This process transforms an algebraic function into a numerical one. Evaluating a function allows us to find specific values of that function for given inputs. This is particularly useful in a variety of practical applications, such as computing values in scientific experiments or financial calculations.
- Given a function, such as \(f(x) = 5x^2 - 3x - 27\), we identify the variable (here, \(x\)).
- We substitute the chosen numerical value for the variable.
- The function is then computed at this specific input, yielding a concrete output.
Polynomial Function
A polynomial function is an expression consisting of variables and coefficients, where the variables have only non-negative integer exponents. These functions can include a wide range of terms. In our exercise, the polynomial function given is quadratic:
- General form: \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\).
- Each term can be identified by its degree, determined by the exponent of its variable.
- Common types include linear (degree 1), quadratic (degree 2), cubic (degree 3), and so forth.
Quadratic Expressions
Quadratic expressions are a specific type of polynomial where the highest exponent of the variable is two. They follow the standard form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
- These expressions often feature prominently in solving problems related to parabolic motion, area calculations, and optimization.
- A quadratic expression results in a symmetrical curve known as a parabola when plotted graphically.
- The leading coefficient, \(a\), influences the direction and width of the parabola.
Substituting Values
Substituting values into an expression is a crucial step in mathematics that initially transforms an algebraic expression into one that's easier to solve or interpret. This is done by replacing the variable in the expression with a given number.
- Ensure precision in substitution to avoid calculation errors.
- Perform operations according to the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).
- It simplifies complex expressions into simpler numeric forms.
