Problem 18 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(h(x)=x^{2}-4 x+5\) a. \(h(4)\) b. \(h(-4)\) c. \(h(0)\)

Step-by-Step Solution

Answer

a. \(h(4) = 5\), b. \(h(-4) = 37\), c. \(h(0) = 5\).

Step 1: Evaluate h(4)

To find \(h(4)\), substitute \(x = 4\) into the function \(h(x)\) which gives \(h(4)= 4^{2}-4(4)+5 = 16 -16 +5 = 5\).

Step 2: Evaluate h(-4)

Next, to evaluate \(h(-4)\), substitute \(x = -4\) into the function \(h(x)\), that gives \(h(-4) = (-4)^{2} - 4(-4) + 5 = 16+16+5 = 37\).

Step 3: Evaluate h(0)

Lastly, to calculate \(h(0)\), we replace \(x = 0\) into the function \(h(x)\) to get \(h(0) = 0 - 4(0) + 5 = 5\).

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial that can be recognized by their standard form: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The term "quadratic" stems from the Latin word "quadratus," meaning "square," referring to the squared term present in these functions. These functions are central in algebra and graphing, showing up in various forms in problems involving area, projectile motion, and more.
The graph of a quadratic function is a curve called a parabola. This parabola can either open upwards, showing a U-shape, or downwards, displaying an upside-down U. The direction in which the parabola opens depends on the sign of the coefficient \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Key characteristics include:

• The vertex, which is the highest or lowest point of the parabola, depending on its direction.
• The axis of symmetry, a vertical line that passes through the vertex, dividing the parabola into two mirror images.
• The y-intercept, which occurs where the graph crosses the y-axis, at the point \((0, c)\).

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions can take different forms, such as monomials (a single term) or polynomials (multiple terms). In the context of our exercise, the function \( h(x) = x^2 - 4x + 5 \) is an algebraic expression representing a quadratic function.
When working with algebraic expressions, it's essential to understand that:

• Variables represent unknown values and can change. In our example, \( x \) is the variable.
• Constants are fixed values that don't change; here, numbers like \( 4 \) and \( 5 \) serve as constants.

Simplifying and evaluating these expressions is crucial, often involving combining like terms or applying operations to arrive at a solution.

Substitution Method

The substitution method is a fundamental technique in algebra used to find the value of a function at a specific point. It involves replacing the variable in the function with a given value.
In our problem, to evaluate the function \( h(x) = x^2 - 4x + 5 \):

• First, identify the value to substitute. In the exercise, we found \( h(4) \), \( h(-4) \), and \( h(0) \).
• Next, replace the variable \( x \) with each given number. This means substituting \( x \) with \( 4 \), \(-4\), or \( 0 \) in our function.
• After substitution, perform the arithmetic operations to simplify and get the result.

This method is helpful because it transforms abstract expressions into concrete calculations, making it easier to understand and solve mathematical problems. It is also widely used across different areas of math, not just quadratics.