Problem 42 - Thinking Mathematically

Question

Evaluate each function at the given value of the variable. \(h(r)=2 r^{2}-4\) a. \(h(5)\) b. \(h(-1)\)

Step-by-Step Solution

Answer

The result of evaluating the function at \(r = 5\) is \(h(5)=46\) and at \(r = -1\) is \(h(-1)= -2\).

Step 1: Evaluate \(h(5)\)

Replace \(r\) in the equation \(h(r)=2r^{2}-4\) with \(5\). So then the equation becomes: \(h(5)= 2(5)^{2}-4\). Solve this computation to get a numerical value.

Step 2: Evaluate \(h(-1)\)

Now replace \(r\) in the equation \(h(r)=2r^{2}-4\) with \(-1\). This new equation is \(h(-1)= 2(-1)^{2}-4\). Also, solve this computation to get another numerical value.

Key Concepts

Algebraic Expressions

In mathematics, an algebraic expression is a combination of constants, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions do not have an equals sign, differentiating them from equations.

For example, in the expression \(2r^2 - 4\), we see:

• **Constants**: These are numbers with fixed value, here 2 and 4.
• **Variables**: The letters that represent unknown values, such as \(r\) in this case.
• **Operations**: The arithmetic operations such as multiplication and subtraction in the expression.

Understanding algebraic expressions is crucial for various branches of mathematics as they form the foundation for equations, inequalities, and functions. By manipulating these expressions according to algebraic rules, one can solve for unknowns or evaluate expressions at specific values of the variables.

Quadratic Functions

Quadratic functions are a type of polynomial function with a degree of 2, meaning the highest power of the variable is squared. The general form of a quadratic function is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

The function \(h(r) = 2r^2 - 4\) is a specific example of a quadratic function where:

• **Leading coefficient**: This is \(a\) which equals 2, determining the shape and direction of the parabola.
• **Constant term**: This is -4, impacting the vertical position of the graph.
• The absence of a linear \(r\) term implies that the vertex of the parabola lies on the y-axis.

These functions graph as parabolas which can open upwards or downwards depending on the sign of \(a\). They are essential in modeling scenarios in physics, finance, and engineering where relationships are nonlinear and symmetric.

Variable Substitution

Variable substitution is a straightforward yet powerful algebraic technique used to simplify expressions and evaluate functions. By replacing a variable in an algebraic expression with a specific number, we can calculate its value for that number.

In the function \(h(r) = 2r^2 - 4\):

• **Substituting \(r=5\)**: Replacing \(r\) with 5, the expression becomes \(h(5) = 2(5)^2 - 4\). Calculating, \(h(5) = 50 - 4 = 46\).
• **Substituting \(r=-1\)**: Similarly, substitute \(r\) with -1 to get \(h(-1) = 2(-1)^2 - 4\). Simplifying, \(h(-1) = 2 - 4 = -2\).

This method allows us to analyze the behavior of functions for different inputs and is fundamental in solving equations and finding roots. It lays a crucial basis for more advanced topics like calculus, where substitution is routinely used with limits and derivatives.