Problem 33 - College Algebra

Question

Graph \(f(x)=-2 x-10 .\) Pick a set of 5 ordered pairs using inputs \(x=-2,1,5,6,9\) and use linear regression to verify the function.

Step-by-Step Solution

Answer

The function is verified; it matches the ordered pairs.

Step 1: Calculate Ordered Pairs

We are given the function \( f(x) = -2x - 10 \). To find the ordered pairs, substitute each given \( x \) value into the function. For \( x = -2 \), \( f(-2) = -2(-2) - 10 = 4 - 10 = -6 \). For \( x = 1 \), \( f(1) = -2(1) - 10 = -2 - 10 = -12 \). For \( x = 5 \), \( f(5) = -2(5) - 10 = -10 - 10 = -20 \). For \( x = 6 \), \( f(6) = -2(6) - 10 = -12 - 10 = -22 \). For \( x = 9 \), \( f(9) = -2(9) - 10 = -18 - 10 = -28 \).The ordered pairs are \((-2, -6), (1, -12), (5, -20), (6, -22), (9, -28)\).

Step 2: Plot the Ordered Pairs

On a coordinate plane, plot each of the ordered pairs: 1. \((-2, -6)\)2. \((1, -12)\)3. \((5, -20)\)4. \((6, -22)\)5. \((9, -28)\)These points should form a straight line, as the equation is linear.

Step 3: Verify Using Linear Regression

Linear regression checks whether the plotted points lie on the line described by \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. With \( m = -2 \) and \( c = -10 \), substitute each x-value into the equation to confirm they provide the corresponding y-values. Each point matches the calculated y-values, confirming the regression.

Key Concepts

Graphing Equations

Graphing equations, especially linear ones, is an essential skill in algebra and helps us visually understand the relationship between variables. When you graph a linear equation like \( f(x) = -2x - 10 \), you represent it as a straight line on a coordinate plane. This relationship is characterized by its slope and y-intercept.

Here's how you do it:

• First, recognize the formula of a linear equation \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
• For our equation, \( m = -2 \) and \( c = -10 \). This tells us the line decreases by 2 for every increase of 1 in \( x \).
• Plotting involves finding points or "ordered pairs" that satisfy the equation. Once you find these points, mark them on the coordinate plane.
• Draw a line that passes through these points. Because the equation is linear, these points will align in a straight line.

Graphing allows us to see a visual representation of the equation and understand patterns or trends in data, which is crucial for further analyses like predictions or decision making.

Ordered Pairs

Ordered pairs are fundamental in understanding how variables relate to each other in equations. An ordered pair consists of an \( x \) coordinate and a \( y \) coordinate, written as \((x, y)\). These pairs help us identify specific points on the graph of an equation.

Here's how you generate ordered pairs:

• Select a series of \( x \) values. In our exercise, we used \( x = -2, 1, 5, 6, 9 \).
• Substitute each \( x \) value into the equation \( f(x) = -2x - 10 \) to calculate the corresponding \( y \) values.
• For example, when \( x = 1 \), \( f(1) = -2(1) - 10 = -12 \). Thus, the ordered pair is \( (1, -12) \).
• Repeat this process for each \( x \) value to create a complete set of ordered pairs: \((-2, -6), (1, -12), (5, -20), (6, -22), (9, -28)\).

Ordered pairs are extremely useful for plotting graphs and analyzing relationships in mathematical functions. They are the foundation of graphing, enabling a visual approach to solving and verifying mathematical equations.

Linear Regression

Linear regression is a powerful statistical tool used to confirm the relationship between two variables, typically \( x \) and \( y \), through a linear equation. It helps to model the data by fitting it to a linear equation and is essential in confirming that the plotted graph represents the equation provided.

Here's how linear regression works in our context:

• The linear equation is of the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
• For our equation \( f(x) = -2x - 10 \), we have \( m = -2 \) and \( c = -10 \).
• Linear regression involves substituting the \( x \)-values into this equation to verify that the calculated \( y \)-values match the ones from your ordered pairs.
• If each ordered pair\((x, y)\) fits this equation, then the graph aligns with the regression, confirming the relationship.

Linear regression is not only a verifying tool but also helps in forecasting and making predictions based on observed data patterns. It's widely used in statistics and economics for identifying trends and relationships in data.