Problem 4 - Algebra for College Students

Question

If \(g(x)=-x^{2}-4 x+6\), find \(g(0), g(5)\). and \(g(-a)\).

Step-by-Step Solution

Answer

\(g(0) = 6\), \(g(5) = -39\), \(g(-a) = -a^2 + 4a + 6\).

Step 1: Substitute and Solve for g(0)

To find the value of \(g(0)\), substitute \(x = 0\) into the equation \(g(x) = -x^2 - 4x + 6\). This gives \(g(0) = -(0)^2 - 4(0) + 6 = 6\).

Step 2: Substitute and Solve for g(5)

To find the value of \(g(5)\), substitute \(x = 5\) into the equation \(g(x) = -x^2 - 4x + 6\). This gives \(g(5) = -(5)^2 - 4(5) + 6 = -25 - 20 + 6 = -39\).

Step 3: Substitute and Solve for g(-a)

To find the expression for \(g(-a)\), substitute \(x = -a\) into the equation \(g(x) = -x^2 - 4x + 6\). This gives \(g(-a) = -(-a)^2 - 4(-a) + 6 = -a^2 + 4a + 6\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function with the highest degree of 2, and its general form is given by \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants where \( a eq 0 \). In our example, the quadratic function is \( g(x) = -x^2 - 4x + 6 \), which means:

• The coefficient \( a = -1 \), indicating the parabola opens downwards.
• The coefficient \( b = -4 \), affects the symmetry and the vertex location.
• The constant term \( c = 6 \), determines the y-intercept of the graph.

Quadratic functions can have their graphs represented as parabolas. Understanding the parts of a quadratic function helps in graphing the equation, predicting the function’s behavior, and solving for specific values.

Substitution Method

The substitution method involves replacing variables with specific values to simplify equations and find results. This is particularly useful in function evaluation. For example, when you have a function like \( g(x) = -x^2 - 4x + 6 \), you can substitute different values of \( x \) to find the function's output.

• To find \( g(0) \), replace \( x \) with 0: \( g(0) = -(0)^2 - 4(0) + 6 = 6 \).
• For \( g(5) \), substitute 5: \( g(5) = -(5)^2 - 4(5) + 6 = -39 \).
• When given \( g(-a) \), use \( -a \) as the substitution: \( g(-a) = -(-a)^2 - 4(-a) + 6 = -a^2 + 4a + 6 \).

By substituting these values, you simplify the expressions and obtain evaluations. This method is key for computing specific outputs of functions in algebra and calculus.

Function Notation

Function notation is a way to express the relationship between inputs and outputs in a mathematical function. Using the format \( f(x) \), we indicate that \( f \) is a function of \( x \). This notation is both flexible and convenient for denoting complex relationships or operations.
For instance, when expressing \( g(x) = -x^2 - 4x + 6 \), the letter \( g \) labels the function, and \( x \) is the input variable. This notation allows us to easily substitute specific values for \( x \) to find corresponding outputs.

• \( g(0) \) represents the output when the input is 0.
• \( g(5) \) indicates substitution of 5 as the input.
• \( g(-a) \) shows a more complex input substitution of \(-a\).

Understanding function notation is crucial for communicating mathematical functions and ensures clarity in problem-solving, helping avoid confusion when dealing with various functions.