Problem 11 - Algebra Form and Function

Question

Without a calculator, match the equations (a)-(g) to the graphs ( \(\mathrm{I}\) ) -(VII). (a) \(y=x-3\) (b) \(-3 x+2=y\) (c) \(2=y\) (d) \(y=-4 x-3\) (e) \(y=x+2\) (f) \(y=x / 3\) (g) \(\quad 4=x\)

Step-by-Step Solution

Answer

Question: Match the equation with its corresponding graph: (a) \(y=x-3\) (d) \(y=-4x-3\) (e) \(y=x+2\) (f) \(y=\frac{1}{3}x\) (b) \(-3x+2=y\) (c) \(2=y\) (g) \(4=x\) Options: (I) A graph with a line rising to the right, crossing the y-axis at -3 (II) A graph with a line that has a steep downward slope and crosses the y-axis at -3 (III) A graph with a line rising to the right, crossing the y-axis at 2 (IV) A graph with a line rising to the right, crossing the y-axis at 0, and less steep than (a) or (e) (V) A graph with a diagonal line crossing the x-axis at 2/3 (between 0 and 1) and crossing the y-axis at 2 (VI) A graph with a horizontal line at \(y=2\) (VII) A graph with a vertical line at \(x=4\) Answer: (a) - (I) (d) - (II) (e) - (III) (f) - (IV) (b) - (V) (c) - (VI) (g) - (VII)

Step 1: Identify the Slope and y-Intercept of Equations in the Form y=mx+b

First,let's identify the equations that are in the form \(y=mx+b\). Equations (a), (d), (e), and (f) can be written in this format. The slope, m, determines the steepness of the line (rise/run), and the positive or negative sign of the slope impacts its direction (positive slopes rise to the right, while negative slopes fall to the right). The y-intercept, b, tells us the point where the line intersects the y-axis: (a) \(y=x-3\): Slope, m=1; y-intercept, b=-3 (d) \(y=-4x-3\): Slope, m=-4; y-intercept, b=-3 (e) \(y=x+2\): Slope, m=1; y-intercept, b=2 (f) \(y=\frac{1}{3}x\): Slope, m=1/3; y-intercept, b=0

Step 2: Identify x-Intercept and y-Intercept for Non y=mx+b Equations

Next, let's analyze the remaining equations (b), (c), and (g). These equations aren't in the form of \(y=mx+b\), but we can still find their x-intercept and y-intercept: (b) \(-3x+2=y\) => \(x=\frac{2-y}{3}\) => x-intercept occurs when y=0 (x=2/3), y-intercept occurs when x=0 (\(y=2\)) (c) \(2=y\): This is a horizontal line, passing through the y-axis at the point where \(y=2\). (g) \(4=x\): This is a vertical line, passing through the x-axis at the point where \(x=4\).

Step 3: Match the Equations with Graphs

Now we can match each equation to its corresponding graph by analyzing their features: (a) \(y=x-3\): Has a positive slope and y-intercept of -3. Look for a graph with a line rising to the right, crossing the y-axis at -3. (d) \(y=-4x-3\): Has a steep negative slope and y-intercept of -3. Look for a graph with a line that has a steep downward slope and crosses the y-axis at -3. (e) \(y=x+2\): Has a positive slope and y-intercept of 2. Look for a graph with a line rising to the right, crossing the y-axis at 2. (f) \(y=\frac{1}{3}x\): Has a positive slope (but less steep than (a) or (e)) and y-intercept of 0. Look for a graph with a line rising to the right, crossing the y-axis at 0, and less steep than (a) or (e). (b) \(-3x+2=y\): This equation has an x-intercept at 2/3 and y-intercept at 2. Look for a graph with a diagonal line crossing the x-axis at 2/3 (between 0 and 1) and crossing the y-axis at 2. (c) \(2=y\): This equation represents a horizontal line passing through the y-axis at 2. Look for a graph with a horizontal line at \(y=2\). (g) \(4=x\): This equation represents a vertical line passing through the x-axis at 4. Look for a graph with a vertical line at \(x=4\).

Key Concepts

Linear Equations

Linear equations are mathematical statements where the highest power of the variable is 1. They can be presented in various forms, but the most common one is the slope-intercept form. The general representation of a linear equation is:

• Standard Form: \(Ax + By = C\)
• Slope-Intercept Form: \(y = mx + b\)

Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept. Linear equations graph as straight lines and define a relationship where a change in the variable \(x\) results in a proportional change in \(y\). This exercise focuses on identifying and graphing these equations to understand their graphical representation better.

Slope-Intercept Form

The slope-intercept form of a linear equation, \(y = mx + b\), is very useful for quickly identifying key features of a line.

• Slope (\(m\)): This value indicates the line's steepness and direction. A positive slope means the line rises as it moves right, while a negative slope means it falls.
• y-Intercept (\(b\)): This is the point where the line crosses the y-axis. It indicates the value of \(y\) when \(x\) is zero.

For example, in the equation \(y = x - 3\), the slope is 1 (since the line rises equally with each step right), and it crosses the y-axis at -3. This form allows for straightforward graphing and makes it easier to interpret the line's behavior visually.

Graph Interpretation

Interpreting graphs involves understanding how linear equations translate into visual data. When looking at a graph:- Identify the slope by assessing the angle of the line. A steeper angle corresponds to a higher absolute value slope.- Determine the y-intercept, where the line meets the y-axis. This can often help confirm your graph corresponds correctly with the equation.- Examine intercepts other than the y-intercept, like the x-intercept. For instance, if you have an equation in the form \(Ax + By = C\), rearranging it to slope-intercept form can be insightful.Analyzing these elements on a graph helps to visualize data relationships and make predictions about the behavior of different variables.

Horizontal and Vertical Lines

Horizontal and vertical lines are special cases of linear equations that represent constant values. - **Horizontal Lines:** These are expressed as \(y = c\), where \(c\) is a constant. The line runs parallel to the x-axis. For example, the equation \(y = 2\) describes a horizontal line that goes through the y-axis at \(y = 2\). There is no slope because there is no change in \(y\) as \(x\) changes.- **Vertical Lines:** These are represented as \(x = c\), where \(c\) stands as a constant. The line is parallel to the y-axis. For instance, \(x = 4\) designates a vertical line crossing the x-axis at \(x = 4\). Vertical lines have an undefined slope as they represent an infinite change in \(y\) for no change in \(x\).Understanding these lines helps in graphing linear equations and recognizing patterns in data relationships.