Problem 22
Question
Determine the slope \(m\) and \(y\) intercept \(b\) of the line with the given equation. Then sketch the line. \(y=2 x-3\)
Step-by-Step Solution
Verified Answer
Slope \(m\) is 2, y-intercept \(b\) is -3. Graph is a straight line through (0, -3) and (1, -1).
1Step 1: Identify the Slope and Y-intercept
The equation of the line is given in the slope-intercept form, which is \( y = mx + b \). In this equation, \(m\) is the slope and \(b\) is the y-intercept. For the given line \(y = 2x - 3\), the slope \(m\) is 2 and the y-intercept \(b\) is -3.
2Step 2: Plot the Y-intercept
On the graph, locate the y-axis, and plot the y-intercept \(b = -3\). This is the point (0, -3) on the graph.
3Step 3: Use the Slope to Determine Another Point
Since the slope \(m = 2\) is a fraction \(\frac{2}{1}\), it means 'rise over run'. From the y-intercept (0, -3), move up 2 units and right 1 unit to plot the next point at (1, -1).
4Step 4: Draw the Line
Connect the plotted points (0, -3) and (1, -1) with a straight line. This line represents the equation \(y = 2x - 3\).
Key Concepts
Equation of a LineSlope CalculationY-intercept Determination
Equation of a Line
When talking about the equation of a line, it's often useful to start with the slope-intercept form. This form is very friendly to use and recognize, especially when you're starting out. In mathematical terms, it's written as \( y = mx + b \). Here, \( y \) and \( x \) are variables for our coordinates on the graph. The value \( m \) represents the slope of the line, and \( b \) is the y-intercept. This format makes it super easy to identify key features of the line from a quick glance.
To better understand, we can look at our exercise example: \( y = 2x - 3 \). This is already in slope-intercept form, allowing us to immediately see that the slope \( m \) is 2 and the y-intercept \( b \) is -3. This form is fantastic because it simplifies both graphing and understanding the relationship between \( x \) and \( y \) as linear. Lines in this form are always straight, making calculations and predictions simpler.
To better understand, we can look at our exercise example: \( y = 2x - 3 \). This is already in slope-intercept form, allowing us to immediately see that the slope \( m \) is 2 and the y-intercept \( b \) is -3. This form is fantastic because it simplifies both graphing and understanding the relationship between \( x \) and \( y \) as linear. Lines in this form are always straight, making calculations and predictions simpler.
Slope Calculation
The slope \( m \) of a line is crucial because it tells us how steep the line is. In simple terms, the slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.
In the context of our equation \( y = 2x - 3 \), the slope \( m \) is 2. But what does that mean? It means for every 1 unit you move to the right on the x-axis, the line goes up by 2 units on the y-axis. Mathematically, this can be seen as \( m = \frac{2}{1} \), confirming that the rise is 2 and the run is 1.
Understanding the slope not only helps in graphing but also in determining how changes in \( x \) affect \( y \). The steeper the slope, the faster \( y \) changes with \( x \). A positive slope like 2 indicates that as \( x \) increases, \( y \) also increases.
In the context of our equation \( y = 2x - 3 \), the slope \( m \) is 2. But what does that mean? It means for every 1 unit you move to the right on the x-axis, the line goes up by 2 units on the y-axis. Mathematically, this can be seen as \( m = \frac{2}{1} \), confirming that the rise is 2 and the run is 1.
Understanding the slope not only helps in graphing but also in determining how changes in \( x \) affect \( y \). The steeper the slope, the faster \( y \) changes with \( x \). A positive slope like 2 indicates that as \( x \) increases, \( y \) also increases.
Y-intercept Determination
The y-intercept of a line is a fundamental concept in graphing. It’s where the line crosses the y-axis, indicating the value of \( y \) when \( x \) is 0. This makes it an easy starting point when drawing the line.
In our example equation \( y = 2x - 3 \), the y-intercept \( b \) is -3. This means the line passes through the point (0, -3) on the graph. You simply locate -3 on the y-axis to mark this point. This is essential, because once you have the y-intercept, you can use the slope to find additional points on the line.
Recognizing the y-intercept not only aids in sketching the line but also gives insight into the conditions of a linear equation, especially in practical scenarios where \( x \) represents a real-world variable and \( y \) the outcome.
In our example equation \( y = 2x - 3 \), the y-intercept \( b \) is -3. This means the line passes through the point (0, -3) on the graph. You simply locate -3 on the y-axis to mark this point. This is essential, because once you have the y-intercept, you can use the slope to find additional points on the line.
Recognizing the y-intercept not only aids in sketching the line but also gives insight into the conditions of a linear equation, especially in practical scenarios where \( x \) represents a real-world variable and \( y \) the outcome.
Other exercises in this chapter
Problem 21
Find the domain and rule of \(g \circ f\) and \(f \circ g\). \(f(x)=x^{2}\) and \(g(x)=\sqrt{x}\)
View solution Problem 22
Find the error. $$ \frac{e^{3}}{e^{\sqrt{2}} e^{\sqrt{2}}} \stackrel{?}{=} \frac{e^{3}}{e^{\sqrt{2}-\sqrt{2}}} \stackrel{?}{=} \frac{e^{3}}{e^{2}} \stackrel{?}{
View solution Problem 22
Sketch the graph of the function. Indicate any intercepts and symmetry, and determine whether the function is even, odd, or neither. $$ 2-\csc x $$
View solution Problem 22
Sketch the graph. List the intercepts and describe the symmetry (if any) of the graph. $$ 4 x^{2}+4 y^{2}=9 $$
View solution