Problem 70 - College Algebra

Question

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2)\). $$ f(x)=8 x^{2}-7 x+3 $$

Step-by-Step Solution

Answer

The function values are: 49, 18, 3, 4, and 21 for \(x = -2, -1, 0, 1, \) and \(2\) respectively.

Step 1: Evaluate f(-2)

Start by substituting \(x = -2\) into the function \(f(x) = 8x^2 - 7x + 3\). \[f(-2) = 8(-2)^2 - 7(-2) + 3\] Calculate:\[8(-2)^2 = 8 \times 4 = 32,\]\[-7(-2) = 14,\]\[32 + 14 + 3 = 49.\] Therefore, \( f(-2) = 49 \).

Step 2: Evaluate f(-1)

Substitute \(x = -1\) into the function.\[f(-1) = 8(-1)^2 - 7(-1) + 3\]Calculate:\[8(-1)^2 = 8 \times 1 = 8,\]\[-7(-1) = 7,\]\[8 + 7 + 3 = 18.\]Therefore, \( f(-1) = 18 \).

Step 3: Evaluate f(0)

Substitute \(x = 0\) into the function.\[f(0) = 8(0)^2 - 7(0) + 3\] This simplifies to:\[0 - 0 + 3 = 3.\] Therefore, \( f(0) = 3 \).

Step 4: Evaluate f(1)

Substitute \(x = 1\) into the function.\[f(1) = 8(1)^2 - 7(1) + 3\]Calculate:\[8(1)^2 = 8 \times 1 = 8,\]\[-7(1) = -7,\]\[8 - 7 + 3 = 4.\]Therefore, \( f(1) = 4 \).

Step 5: Evaluate f(2)

Substitute \(x = 2\) into the function.\[f(2) = 8(2)^2 - 7(2) + 3\]Calculate:\[8(2)^2 = 8 \times 4 = 32,\]\[-7(2) = -14,\]\[32 - 14 + 3 = 21.\]Therefore, \( f(2) = 21 \).

Key Concepts

Quadratic Function

A quadratic function is a type of function that can be written in the form \( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The term with the highest power of \(x\) is \(x^2\). This is why it is called quadratic. Quadratic functions graph as a parabola, which is a u-shaped curve. This curve can open upwards or downwards, depending on the sign of \(a\). For example, the function given in the exercise, \( f(x) = 8x^2 - 7x + 3 \), is a quadratic function. Here, \(a = 8\), \(b = -7\), and \(c = 3\). Since \(a\) is positive, the parabola opens upwards. Understanding the structure of a quadratic function helps us to evaluate it more effectively.

Substitution Process

The substitution process refers to replacing the variable \( x \) in a function with a specific value, allowing us to find the function's output at that point. This process is essential when evaluating functions, like the quadratic function in the exercise.
To substitute, follow these simple steps:

• Select the value you wish to substitute into the function.
• Replace every instance of \( x \) with this value in the expression.
• Carry out any necessary arithmetic to solve the expression.

For example, to find \( f(-2) \) for the given function \( f(x) = 8x^2 - 7x + 3 \), substitute \(-2\) for \( x \) to get \( 8(-2)^2 - 7(-2) + 3 \), then calculate the result. This approach is systematic and can be applied to any position on the function's graph where you need to know the value.

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial function for a particular value of \( x \). Polynomials, like the quadratic function in the exercise, consist of terms that are powers of \( x \) with coefficients. In the case of our exercise's quadratic function, each term's contribution is computed and summed together.
Evaluating polynomials follows these steps:

• Compute each term separately by substituting \( x \) with the required value.
• Perform the arithmetic operations like multiplication of terms, addition, or subtraction.
• Sum up the results of each term to find the final value of the polynomial.

For instance, with \( f(1) = 8(1)^2 - 7(1) + 3 \), first calculate \( 8 \times 1^2 = 8 \), then \(- 7 \times 1 = -7\), then sum to find \( 8 - 7 + 3 = 4 \). Mastering polynomial evaluation is key for understanding a wide range of mathematical problems.

Step-by-Step Solution

A step-by-step solution is a clear breakdown of solving a problem, which can be particularly useful in learning and understanding complex problems such as function evaluations. In this context, a step-by-step approach involves:

• Listing each calculation clearly and methodically.
• Taking one step at a time without skipping any part of the process.
• Providing the reasoning or rules behind each operation or substitution.

Using the quadratic function \( f(x) = 8x^2 - 7x + 3 \), the step-by-step solution allows us to see how each value of \( x \) translates into a calculated outcome. For example, substituting \( x = 0 \) immediately simplifies the terms significantly, showing its importance. Step-by-step solutions help demystify complex problems, turning them into manageable, bite-sized tasks. Such clarity promotes better understanding and learning efficiency.