Problem 17 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(h(x)=x^{2}-2 x+3\) a. \(h(4)\) b. \(h(-4)\) c. \(h(0)\)

Step-by-Step Solution

Answer

The results are: \(h(4) = 11\), \(h(-4) = 27\) and \(h(0) = 3\).

Step 1: Evaluate \(h(4)\)

Substitute \(x=4\) into the function \(h(x)=x^{2}-2 x+3\). It gives you \(4^{2}-2*4+3=16-8+3=11\). Therefore, \(h(4) = 11\).

Step 2: Evaluate \(h(-4)\)

Substitute \(x=-4\) into the function \(h(x)=x^{2}-2 x+3\). It gives you \((-4)^{2}-2*(-4)+3=16+8+3=27\). Therefore, \(h(-4) = 27\).

Step 3: Evaluate \(h(0)\)

Substitute \(x=0\) into the function \(h(x)=x^{2}-2 x+3\). It gives you \(0^{2}-2*0+3=0-0+3=3\). Therefore, \(h(0) = 3\).

Key Concepts

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be represented in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. These functions graph as parabolas in the coordinate plane, with the highest exponent of the variable \( x \) being 2.
Understanding the shape and position of the parabola involves analyzing the coefficients. The coefficient \( a \) determines whether the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)). The vertex form of a quadratic function gives insights into the vertex, which is the highest or lowest point on the graph.
A quadratic function like \( h(x) = x^{2} - 2x + 3 \) is in standard form and tells us initial information about its graph. The symmetry and shape of these graphs make quadratic functions useful for modeling various real-world situations.

Substitution in Algebra

Substitution is a fundamental method in algebra used for determining function values by replacing the variable with a specified number. This concept is crucial for evaluating functions, like the quadratic function in our example.
When tasked with evaluating \( h(x) = x^{2} - 2x + 3 \) for a specific \( x \), substitution involves plugging that specific value into the function.

• For \( h(4) \), replace \( x \) with 4 to find \( 16 - 8 + 3 = 11 \).
• For \( h(-4) \), substitute \( x = -4 \) resulting in \( 16 + 8 + 3 = 27 \).
• For \( h(0) \), substituting \( x = 0 \) simplifies to \( 3 \).

Substitution helps in simplifying expressions and provides the exact value of a function at specific points. It's like finding the output of a machine for a given input.

Working with Algebraic Expressions

Algebraic expressions consist of variables, constants, and arithmetic operations. They form the building blocks of mathematical formulas and functions like our quadratic function.
In the function \( h(x) = x^{2} - 2x + 3 \), the algebraic expression allows us to perform calculations for any value of \( x \). Understanding how to manipulate these expressions is key to solving equations and evaluating functions.

• Exponents show repeated multiplication, so \( x^2 \) means \( x \) multiplied by itself.
• Constants like \( -2x \) indicate a fixed value that modifies \( x \).
• Knowing the order of operations—parentheses, exponents, multiplication and division, addition, and subtraction (PEMDAS)—is crucial to accurately simplifying expressions.

Mastering algebraic expressions enables students to analyze more complex algebraic structures and solve a wide variety of mathematical problems efficiently.