Problem 29 - A Graphical Approach to College Algebra

Question

Concept Check The function \(\mathrm{Y}_{2}\) is defined as \(\mathrm{Y}_{1}+k\) for some real number \(k\). Based on the two views of the graphs of \(\mathrm{Y}_{1}\) and \(\mathrm{Y}_{2}\) and the displays at the bottoms of the screens, what is the value of \(k ?\) \((6,2)\) lies on the graph of \(Y_{1}\) First view (Graph can't copy) \((6,-1)\) lies on the graph of \(Y_{2}\) Second view (Graph can't copy)

Step-by-Step Solution

Answer

The value of \( k \) is \(-3\).

Step 1: Understand the Problem

We are given that \( Y_2 = Y_1 + k \) where \( k \) is a real number. We need to find the value of \( k \). Additionally, two points are provided: \( (6,2) \) is on \( Y_1 \) and \( (6,-1) \) is on \( Y_2 \).

Step 2: Use Given Points

We know that \( (6,2) \) is on the graph of \( Y_1 \), meaning \( Y_1(6) = 2 \). Similarly, \( (6,-1) \) is on the graph of \( Y_2 \), meaning \( Y_2(6) = -1 \).

Step 3: Apply the Definition of Y2

According to the problem, \( Y_2 = Y_1 + k \). Substitute the known values for \( Y_1(6) \) and \( Y_2(6) \): \( -1 = 2 + k \).

Step 4: Solve for k

To find \( k \), we solve the equation: \( -1 = 2 + k \). Subtract 2 from both sides to get \( k = -3 \).

Key Concepts

Linear Equations

A linear equation is an algebraic equation that models a straight line when plotted on a graph. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept of the line. Linear equations are fundamental in coordinate geometry as they establish a relationship between two variables, usually \(x\) and \(y\). These types of equations help in predicting one variable's outcome with respect to another.

• **Slope (\(m\))**: Represents the rate at which \(y\) changes for a change in \(x\). A positive slope means the line ascends, while a negative slope descends.
• **Y-intercept (\(b\))**: Is the point where the line crosses the y-axis, i.e., where \(x = 0\).

Understanding linear equations is crucial for solving many mathematical problems, including transformation of graphs, which involves shifting, reflecting, or stretching lines on a graph by changing parameters like \(m\) or \(b\). In the context of our exercise, the transformation involves adding a constant \(k\) to each output value of a given linear function, illustrating a vertical shift of the graph.

Function Notation

Function notation is a way to express a function's operations and outputs clearly and concisely. It allows us to use letters such as \(f(x)\) or \(Y(x)\) to denote functions and clarify that an input \(x\) corresponds to a particular output.
Function notation is especially helpful in problems requiring numerous evaluations of functions, such as the one in our exercise. It's essential to grasp the notation:

• **General form**: \(f(x) = \, expression\). For instance, \(f(x) = x + 3\) suggests an expression dependent on the variable \(x\).
• **Evaluating functions**: Replace \(x\) with any number or expression to calculate the result. For example, if \(f(x) = x + 3\), then \(f(2) = 2 + 3 = 5\).

In the exercise, we label two functions \(Y_1\) and \(Y_2\) that depend on the same variable but differ by a constant \(k\). This notation design shows how one function is derived from another, emphasizing the utility of consistent function notation in manipulating and comparing functions.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, unites algebra and geometry using graphs. It represents geometric shapes and lines within a coordinate plane using algebraic equations. One key aspect of coordinate geometry is plotting points using ordered pairs \((x, y)\), which indicates a specific location on the graph. Coordinate geometry helps solve problems and understand relations among geometric figures by using:

• **Points**: As seen in our exercise with points \((6,2)\) and \((6,-1)\). Each point indicates a location of the function's graph in the \(xy\)-plane.
• **Graphs**: Visual representations of equations using the coordinate plane. Lines, curves, and shapes can all be presented graphically.

In this particular exercise, we use points to determine how a transformation has shifted the graph of one function to another by observing the changes in their corresponding charts. This empowers students to visualize and solve equations effectively, proving graphs are more than mere representations—they're powerful problem-solving tools.