Problem 41 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \(g(t) = 4t^2-3t+5\) (a) \(g(2)\) (b) \(g(t-2)\) (c) \(g(t)-g(2)\)

Step-by-Step Solution

Answer

\[g(2) = 15, g(t-2) = 4t^2-19t+27, g(t)-g(2) = 4t^2 -3t -10\]

Step 1: Evaluate \(g(2)\)

Substitute \(t=2\) into the function \(g(t)\): \[g(2) = 4(2)^2 - 3(2)+5 = 16 - 6 + 5 = 15\]

Step 2: Evaluate \(g(t-2)\)

Substitute \(t-2\) into the function \(g(t)\): \[g(t-2) = 4(t-2)^2 - 3(t-2)+5 = 4(t^2-4t+4) - 3t+6+5 = 4t^2 -16t +16 -3t+6+5 = 4t^2-19t+27\]

Step 3: Evaluate \(g(t)-g(2)\)

Substitute \(g(t) = 4t^2-3t+5\) and \(g(2) = 15\) into \(g(t)-g(2)\): \[g(t)-g(2) = (4t^2-3t+5) - 15 = 4t^2 -3t -10\]

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions that describe a parabolic curve on a graph. These functions are typically presented in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The leading term \(ax^2\) determines the direction and width of the parabola's opening. Quadratic functions are essential in a variety of real-world contexts, such as physics and engineering, because of their ability to model scenarios like projectile motion.
In this exercise, we are tasked with evaluating the quadratic function \(g(t) = 4t^2 - 3t + 5\). Understanding how the function behaves is critical when we substitute different values for \(t\), as it changes the position or shape of the parabola. This task adds depth to the understanding of quadratic relationships and their implications.

Substitution Method

The substitution method is an invaluable technique used to evaluate functions at specific points. It involves replacing the variable with a given number or expression to find the function's value at that point.
This exercise demonstrates the substitution method in several scenarios:

• Substituting \(t = 2\) into the function \(g(t)\). This was a straightforward calculation that resulted in a specific numerical value \(g(2) = 15\).
• Substituting \(t - 2\) into the function. This is a bit more complex because it requires substituting an expression, not just a number. The result, \(g(t-2) = 4t^2 - 19t + 27\), is a new quadratic expression.

By practicing the substitution method with both numbers and expressions, students strengthen their problem-solving skills and deepen their understanding of how functions work.

Simplifying Expressions

Simplifying expressions is a critical step in evaluating functions. It is the process of reducing expressions to their simplest form. This can involve combining like terms, reducing coefficients, or even factoring if necessary.
In the solved exercise:

• Simplification was key when evaluating \(g(t-2)\) to transform it into \(4t^2 - 19t + 27\). Distributing and combining terms were necessary to arrive at this form.
• When calculating \(g(t) - g(2)\), simplifying resulted in the expression \(4t^2 - 3t - 10\). This involved combining results from prior evaluations.

By practicing simplifying expressions, students can handle more complicated expressions effectively, leading to clearer and correct solutions.