Problem 28 - College Algebra

Question

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). $$ f(x)=-5 x^{2}+2 x-1 $$

Step-by-Step Solution

Answer

\(f(-3) = -52\), \(f(2) = -17\), \(f(-a) = -5a^2 - 2a - 1\), \(-f(a) = 5a^2 - 2a + 1\), \(f(a+h) = -5a^2 - 10ah - 5h^2 + 2a + 2h - 1\).

Step 1: Substitute -3 into f(x)

Replace every instance of \(x\) in the function \(f(x) = -5x^2 + 2x - 1\) with \(-3\). Perform the calculations: \(f(-3) = -5(-3)^2 + 2(-3) - 1 = -5(9) - 6 - 1 = -45 - 6 - 1 = -52\). Thus, \(f(-3) = -52\).

Step 2: Substitute 2 into f(x)

Replace every instance of \(x\) in the function with \(2\). Calculate \(f(2) = -5(2)^2 + 2(2) - 1 = -5(4) + 4 - 1 = -20 + 4 - 1 = -17\). Thus, \(f(2) = -17\).

Step 3: Substitute -a into f(x)

Replace every instance of \(x\) in the function with \(-a\). Simplify to get \(f(-a) = -5(-a)^2 + 2(-a) - 1 = -5a^2 - 2a - 1\). Thus, \(f(-a) = -5a^2 - 2a - 1\).

Step 4: Calculate -f(a)

First find \(f(a)\) by substituting \(a\) into the function: \(f(a) = -5a^2 + 2a - 1\). Therefore, \(-f(a) = -(-5a^2 + 2a - 1) = 5a^2 - 2a + 1\). Thus, \(-f(a) = 5a^2 - 2a + 1\).

Step 5: Substitute a+h into f(x)

Replace every instance of \(x\) in the function with \(a+h\). Calculate \(f(a+h) = -5(a+h)^2 + 2(a+h) - 1\). Use the expansion \((a+h)^2 = a^2 + 2ah + h^2\), then expand: \(-5(a^2 + 2ah + h^2) + 2a + 2h - 1 = -5a^2 - 10ah - 5h^2 + 2a + 2h - 1\). Thus, \(f(a+h) = -5a^2 - 10ah - 5h^2 + 2a + 2h - 1\).

Key Concepts

Quadratic Functions

Quadratic functions are expressions that take the form \( ax^2 + bx + c \). These functions are called "quadratic" because the highest exponent of the variable \( x \) is squared. The general shape of a quadratic function is a parabola, which can open upwards or downwards based on the sign of the leading coefficient \( a \).

Key features of quadratic functions include:

• A vertex, which is the highest or lowest point of the parabola.
• A line of symmetry, which passes through the vertex.
• The parabola may have zero, one, or two x-intercepts, where the graph crosses the x-axis.
• A y-intercept, where the graph crosses the y-axis.

In the given function \( f(x) = -5x^2 + 2x - 1 \), the coefficient \(-5\) tells us that the parabola opens downwards. The other coefficients influence the location of the vertex, as well as the graph's stretch or shrink.

Substitution Method

The substitution method is a straightforward technique used to evaluate functions by replacing each variable with the given value. This method is particularly useful when dealing with polynomial expressions like quadratic functions.

For example, to evaluate \( f(-3) \), you substitute \(-3\) for every instance of \( x \) in the function \(-5x^2 + 2x - 1\):

• Begin with \(-5(-3)^2 + 2(-3) - 1\).
• Calculate \(-5\times9 = -45,\) \( 2\times(-3) = -6,\) and combine these with \(-1\).
• This results in \(-45 - 6 - 1 = -52\)

By using this method across various values, you can see how the function behaves and what outputs it produces for different inputs.

Function Notation

Function notation is a way of representing functions in a more convenient form. Instead of a single equation, it uses notation like \( f(x) \) to denote a function of \( x \). This notation not only simplifies the expression but also clearly indicates the dependent relationship.

In function notation:

• \( f(x) \) means "the result of applying function \( f \) to \( x \)."
• To evaluate, substitute the given value into the variable \( x \).
• For example, \( f(2) \) tells us to plug 2 into every occurrence of \( x \) in the function.

This notation is especially useful when dealing with more complex expressions because it provides clarity on which operations depend on the input \( x \). Functions such as \( f(x) \) from our example can generate multiple outputs like \( f(a) \), \( f(-a) \), or \( f(a+h) \), each representing a different possible input substitution.

Polynomial Expressions

Polynomial expressions are combinations of variables and coefficients involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. A polynomial can be simple, like \( 2x + 1 \) or more complex with higher powers like the quadratic \(-5x^2 + 2x - 1\).

Understanding polynomials involves:

• Recognizing the degree of the polynomial, the highest exponent of variable \( x \), which determines the function's shape and complexity.
• Performing operations such as expansion and simplification, often seen in calculations like evaluating \( f(a+h) \).
• Combining terms, which means reorganizing and simplifying expressions by grouping like terms.

Our quadratic function example demonstrates these principles: \(-5x^2 + 2x - 1\) has terms that include squared terms, linear terms, and constant terms. By recognizing such structures, you can apply characteristics unique to polynomials when evaluating or graphing them.