Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

If \(f(x)=-x^{2}-x-4\), find \(f(-2), f(0)\), and \(f(4)\).

Step-by-Step Solution

Answer

\(f(-2) = -6\), \(f(0) = -4\), \(f(4) = -24\).

Step 1: Understand the Function

The function given is \(f(x) = -x^2 - x - 4\). This is a quadratic function, where the coefficients are \(a = -1\), \(b = -1\), and \(c = -4\). We will plug specific values of \(x\) into this function to find \(f(-2), f(0),\) and \(f(4)\).

Step 2: Calculate \(f(-2)\)

Substitute \(x = -2\) into the function: \[f(-2) = -(-2)^2 - (-2) - 4 \]Calculate step by step: \[(-2)^2 = 4 \]\[- (4) = -4 \]\[- (-2) = 2 \]\[f(-2) = -4 + 2 - 4 = -6 \]The computed value for \(f(-2)\) is \(-6\).

Step 3: Calculate \(f(0)\)

Substitute \(x = 0\) into the function: \[f(0) = -(0)^2 - 0 - 4 \]Calculate step by step: \[f(0) = -0 - 0 - 4 = -4 \]The computed value for \(f(0)\) is \(-4\).

Step 4: Calculate \(f(4)\)

Substitute \(x = 4\) into the function: \[f(4) = -(4)^2 - 4 - 4 \]Calculate step by step: \[(4)^2 = 16 \]\[-(16) = -16 \]\[f(4) = -16 - 4 - 4 = -24 \]The computed value for \(f(4)\) is \(-24\).

Key Concepts

Function Evaluation

Function evaluation is a crucial concept in mathematics, especially within the study of quadratic functions. It means finding the output of a function for a particular input value. This involves substituting the given input into the function and performing the necessary calculations to get a result or output.

For example, if we have a function defined as \( f(x) = -x^2 - x - 4 \), we can evaluate it at different values of \( x \) such as \( -2, 0, \) and \( 4 \).

• For \( f(-2) \), substitute \(-2\) into the function to find its output.
• For \( f(0), \) replace \( x \) with \( 0 \) and evaluate.
• For \( f(4), \) use \( 4 \) as the input.

Function evaluation is a systematic process that reflects how a quadratic function behaves at different points, providing insight into its characteristics and graph.

Substitution Method

The substitution method is a straightforward, yet essential technique used when evaluating functions. In this method, you simply replace the variable in the function with the specific values you want to evaluate.

Let's illustrate this with an example using our function \( f(x) = -x^2 - x - 4 \). Suppose you need to find \( f(-2) \). You substitute \( x = -2 \) into the function:

• Start by computing \( (-2)^2 \) which equals \( 4 \).
• Then, multiply by \(-1\) due to the negative sign in front, giving \(-4\).
• Substitute to get \( f(-2) = -4 + 2 - 4 \), simplifying to \(-6\).

This method involves a sequence of arithmetic operations, ensuring you substitute correctly and follow multiplication rules especially with negative numbers.
The substitution method not only helps with function evaluation but also deepens your understanding of how functions transform numbers.

Parabolas

When dealing with quadratic functions such as \( f(x) = -x^2 - x - 4 \), the graphical representation is a parabola. Understanding the structure of parabolas can help us gain insights into the function's behavior and properties.

Quadratic functions typically take the standard form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. The coefficient \( a \) determines the parabola's direction:

• If \( a > 0 \), the parabola opens upward.
• If \( a < 0 \), like our example, it opens downward.

The axis of symmetry of the parabola can be found using the formula \( x = -\frac{b}{2a} \), and the vertex provides either a maximum or minimum point given the parabola’s orientation.
Understanding parabolas is critical as they appear frequently in mathematical problems and real-life applications such as physics trajectories and optimization problems.