Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ y=2 x-3 $$

Step-by-Step Solution

Answer

The x-intercept is \( \left( \frac{3}{2}, 0 \right) \) and the y-intercept is \( (0, -3) \).

Step 1: Identify the Equation Type

The equation provided is a linear equation in the slope-intercept form, which is given by \( y = mx + b \). In this case, the equation is \( y = 2x - 3 \), where \( m = 2 \) is the slope and \( b = -3 \) is the y-intercept.

Step 2: Calculate the Y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation \( y = 2x - 3 \). This gives \( y = 2(0) - 3 = -3 \). Thus, the y-intercept is the point \( (0, -3) \).

Step 3: Calculate the X-Intercept

For the x-intercept, set \( y = 0 \) in the equation \( y = 2x - 3 \). Solving for \( x \), we have \( 0 = 2x - 3 \) which simplifies to \( 2x = 3 \), and thus \( x = \frac{3}{2} \). Therefore, the x-intercept is the point \( \left( \frac{3}{2}, 0 \right) \).

Step 4: Plot the Intercepts

On the coordinate plane, plot the y-intercept \( (0, -3) \) and the x-intercept \( \left( \frac{3}{2}, 0 \right) \).

Step 5: Draw the Line through the Intercepts

Connect the points \( (0, -3) \) and \( \left( \frac{3}{2}, 0 \right) \) with a straight line. This line represents the graph of the equation \( y = 2x - 3 \).

Key Concepts

The X-Intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the value of \( y \) is always zero. To find the x-intercept, you need to set \( y = 0 \) in the equation of the line and solve for \( x \).

For example, in the equation \( y = 2x - 3 \), setting \( y \) to zero results in \( 0 = 2x - 3 \). Solving for \( x \), we get:

• Add 3 to both sides: \( 3 = 2x \)
• Divide by 2: \( x = \frac{3}{2} \)

Thus, the x-intercept is at the point \( \left( \frac{3}{2}, 0 \right) \). This means that the graph will cross the x-axis at \( x = \frac{3}{2} \).

Understanding where the graph crosses the x-axis gives us insight into the solution to the equation when \( y \) is zero.

The Y-Intercept

The y-intercept is the point where the graph of the equation crosses the y-axis. At the y-intercept, the value of \( x \) is zero. To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \).

Using the equation \( y = 2x - 3 \), let's find the y-intercept:

• Plug \( x = 0 \) into the equation: \( y = 2(0) - 3 \)
• Compute \( y \): \( y = -3 \)

This gives us the y-intercept point of \( (0, -3) \).

This point tells us that our graph will cross the y-axis at \( y = -3 \). Finding the y-intercept quickly can help when drawing the line because it provides a precise point through which we know the line will pass.

Understanding Slope-Intercept Form

The slope-intercept form is a straightforward way of writing the equation of a line. The format is \( y = mx + b \), where:

• \( m \) represents the slope of the line
• \( b \) is the y-intercept, or where the line crosses the y-axis

In our given exercise, the equation \( y = 2x - 3 \) is already in slope-intercept form. Here, the slope \( m \) is 2 and the y-intercept \( b \) is -3. The slope is important because it tells us how steep the line is and the direction it goes. A slope of 2 means that for every unit increase in \( x \), \( y \) increases by 2 units.

Knowing the slope-intercept form allows for easy graphing. You can start by plotting the y-intercept and using the slope to find another point. With both the y-intercept and additional points, you can draw the line accurately. This form is particularly useful in quickly sketching graphs and understanding the basic trend of the line.