Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-10: If \(a\) and \(h\) are real numbers, find (a) \(f(a)\) (b) \(f(-a)\) (c) \(-f(a)\) (d) \(f(a+h)\) (e) \(f(a)+f(h)\) (f) \(\frac{f(a+h)-f(a)}{h}\), if \(h \neq 0\) $$ f(x)=2 x^{2}+3 x-7 $$

Step-by-Step Solution

Answer

(a) \(2a^2 + 3a - 7\); (b) \(2a^2 - 3a - 7\); (c) \(-2a^2 - 3a + 7\); (d) \(2a^2 + 4ah + 2h^2 + 3a + 3h - 7\); (e) \(2a^2 + 3a + 2h^2 + 3h - 14\); (f) \(4a + 2h + 3\) if \(h \neq 0\).

Step 1: Evaluate \( f(a) \)

To find \( f(a) \), substitute \( a \) for \( x \) in the function \( f(x) = 2x^2 + 3x - 7 \). So, \( f(a) = 2(a)^2 + 3(a) - 7 \). Simplify to get \( f(a) = 2a^2 + 3a - 7 \).

Step 2: Evaluate \( f(-a) \)

To find \( f(-a) \), substitute \( -a \) for \( x \) in the function. So, \( f(-a) = 2(-a)^2 + 3(-a) - 7 \). Simplify to get \( f(-a) = 2a^2 - 3a - 7 \).

Step 3: Evaluate \( -f(a) \)

To find \( -f(a) \), take the negative of the result of Step 1: \( -f(a) = -(2a^2 + 3a - 7) \). Simplifying this, we have \( -f(a) = -2a^2 - 3a + 7 \).

Step 4: Evaluate \( f(a+h) \)

To find \( f(a+h) \), substitute \( a+h \) for \( x \) in the function. So, \( f(a+h) = 2(a+h)^2 + 3(a+h) - 7 \). Expand the terms: \( f(a+h) = 2(a^2 + 2ah + h^2) + 3a + 3h - 7 \). Simplify to get \( f(a+h) = 2a^2 + 4ah + 2h^2 + 3a + 3h - 7 \).

Step 5: Evaluate \( f(a) + f(h) \)

To find \( f(a) + f(h) \), compute \( f(h) = 2h^2 + 3h - 7 \) and add it to \( f(a) \) from Step 1. So, \( f(a) + f(h) = (2a^2 + 3a - 7) + (2h^2 + 3h - 7) \). Simplify to get \( f(a) + f(h) = 2a^2 + 3a + 2h^2 + 3h - 14 \).

Step 6: Evaluate \( \frac{f(a+h) - f(a)}{h} \), if \( h \neq 0 \)

Using the result from Step 4 for \( f(a+h) \) and Step 1 for \( f(a) \), find the difference \( f(a+h) - f(a) = (2a^2 + 4ah + 2h^2 + 3a + 3h - 7) - (2a^2 + 3a - 7) \). Simplify to \( 4ah + 2h^2 + 3h \). Divide by \( h \) gives \( \frac{4ah + 2h^2 + 3h}{h} = 4a + 2h + 3 \). This simplification is valid if \( h eq 0 \).

Key Concepts

Function Evaluation

Function evaluation is the process of inputting a given value into a function to determine the output. In the context of polynomial functions, this involves substituting the specified variable with a given number or expression. For example, to evaluate the function \( f(x) = 2x^2 + 3x - 7 \) at \( x = a \), you follow these steps:

• Substitute \( x \) with \( a \): The function becomes \( f(a) = 2a^2 + 3a - 7 \).
• Simplify the expression if necessary.

This process can be repeated for any value, and each substitution offers insight into how altering the input affects the output. This is fundamental in understanding the behavior of functions, as knowing how a function operates on a given input helps predict outcomes for other values.

Polynomial Functions

Polynomial functions are a type of mathematical expression that involves variables raised to integer powers and combined using addition, subtraction, or multiplication. Here, the function \( f(x) = 2x^2 + 3x - 7 \) is a quadratic polynomial, characterized by the highest power of the variable \( x \) being two.Key features of polynomial functions include:

• **Degree**: The highest power of the variable. In our function, the degree is 2, making it a quadratic.
• **Coefficients**: The numerical factors of terms, such as 2 and 3 in the expression.
• **Constant terms**: The standalone number without a variable, like \(-7\) here.

Polynomial functions are crucial in various mathematical applications because their graphs, which appear as smooth and continuous lines or curves, help model real-world phenomena. Quadratics, in particular, often describe parabolic trajectories, illustrating everything from a thrown ball to economic trends.

Difference Quotient

The difference quotient is a crucial concept in calculus, representing the average rate of change of a function. It forms the basis for understanding derivatives. For a function \( f(x) \), the difference quotient is typically structured as \( \frac{f(x + h) - f(x)}{h} \), where \( h eq 0 \). This equation indicates how the function changes as the input changes by \( h \).In our example, the function \( f(x) = 2x^2 + 3x - 7 \) evaluates the difference quotient at \( a \) as:

• Calculate \( f(a + h) \) and \( f(a) \).
• Find the difference: \( f(a + h) - f(a) = 4ah + 2h^2 + 3h \).
• Divide by \( h \) (assuming \( h eq 0 \)): The quotient simplifies to \( 4a + 2h + 3 \).

Through the difference quotient, you analyze how function outputs are influenced by small input shifts, paving the path to deeper insights in calculus regarding instantaneous rates of change and derivatives.