Question

introduction:
polynomials: the shape - shifters of algebra! each one has its own style and a special job to do. were going to dig into these different looks and find out what each one is really good for. so, lets dive in and spot all the different forms of polynomials. ready? lets get started!
part i:
the equation of a line can take on various forms, with two commonly used ones being:

• general form, (ax + by + c = 0)
• slope - intercept form, (y = mx + b)

1. lets delve into reviewing and exploring these forms.

a) consider the equations (3x - 2y - 2 = 0) and (y=\frac{3}{2}x - 1). these represent the same line in different formats. click on this link here, which will bring you to an online graphing tool, desmos, where you can confirm they are the same. sketch the line on the cartesian plane below.
(a cartesian plane grid is shown here)
b) key features of a line include its slope, (x) - intercept, and (y) - intercept. identify and label these on the graph. hint: you can obtain this information on the desmos graph by clicking on the points.
c) consider how each form of the equation highlights these features (slope, (x) - intercept, (y) - intercept). which form is easier to spot them with?

Explanation:

Step1: Analyze the Slope-Intercept Form

The slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. For the equation \( y=\frac{3}{2}x - 1 \), we can directly identify:

• Slope (\( m \)): \( \frac{3}{2} \) (this tells us the line rises 3 units for every 2 units it runs to the right).
• \( y \)-intercept (\( b \)): \( - 1 \) (so the line crosses the \( y \)-axis at \( (0, - 1) \)).

Step2: Find the \( x \)-intercept

To find the \( x \)-intercept, we set \( y = 0 \) in either equation. Let's use the slope - intercept form \( y=\frac{3}{2}x - 1 \):

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So the \( x \)-intercept is at \( (\frac{2}{3},0) \).

Step3: Sketch the Line

• Plot the \( y \)-intercept \( (0, - 1) \).
• Use the slope \( \frac{3}{2} \): from \( (0, - 1) \), move up 3 units and right 2 units to get another point (e.g., \( (2,2) \)), or down 3 units and left 2 units to get \( (- 2,-4) \).
• Draw a straight line through these points (and the \( x \)-intercept \( (\frac{2}{3},0) \) which lies between \( (0, - 1) \) and \( (2,2) \)).

Step4: Analyze the General Form for Features

The general form is \( ax+by + c = 0 \) (here \( 3x-2y - 2=0 \)). To find the slope from the general form, we can rearrange it to slope - intercept form:

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To find the \( x \)-intercept from the general form, set \( y = 0 \): \( 3x-2(0)-2 = 0\Rightarrow3x=2\Rightarrow x=\frac{2}{3} \). To find the \( y \)-intercept, set \( x = 0 \): \( 3(0)-2y-2=0\Rightarrow - 2y=2\Rightarrow y=-1 \).

Step5: Compare the Forms for Spotting Features

• For the slope - intercept form (\( y = mx + b \)): The slope (\( m \)) and \( y \)-intercept (\( b \)) are directly visible. We only need to solve for \( x \) when \( y = 0 \) to find the \( x \)-intercept, but the slope and \( y \)-intercept are immediate.
• For the general form (\( ax + by + c = 0 \)): We need to rearrange the equation to find the slope and \( y \)-intercept. So the slope - intercept form is easier to spot the slope and \( y \)-intercept with, and once we know the slope and \( y \)-intercept, finding the \( x \)-intercept is straightforward.

Answer:

a)

The line can be sketched by plotting the \( y \)-intercept \( (0,-1) \), using the slope \( \frac{3}{2} \) to find other points (e.g., \( (2,2) \) or \( (-2,-4) \)) and drawing a straight line through them (also passing through the \( x \)-intercept \( (\frac{2}{3},0) \)).

b)

• Slope: \( \frac{3}{2} \) (the rate of change of \( y \) with respect to \( x \)).
• \( x \)-intercept: \( (\frac{2}{3},0) \) (where the line crosses the \( x \)-axis).
• \( y \)-intercept: \( (0,-1) \) (where the line crosses the \( y \)-axis). These can be labeled on the graph.

c)

The slope - intercept form (\( y=mx + b \)) is easier to spot the slope (since \( m \) is directly given) and the \( y \)-intercept (since \( b \) is directly given). For the \( x \)-intercept, we can still find it by setting \( y = 0 \) in the slope - intercept form, but the initial identification of slope and \( y \)-intercept is more straightforward in slope - intercept form compared to general form.