Problem 11 - Precalculus

Question

The graph of each equation is a straight line. Graph the equation after finding the \(x\)-and the \(y\) -intercepts. (since you are given that the graph is a line, you need only plot two points before drawing the line.) $$y=2 x-4$$

Step-by-Step Solution

Answer

The x-intercept is (2, 0); the y-intercept is (0, -4).

Step 1: Find the x-intercept

The x-intercept is the point where the line crosses the x-axis. This means the y-value is 0. To find the x-intercept, set \( y = 0 \) in the equation \( y = 2x - 4 \): \[ 0 = 2x - 4 \]Add 4 to both sides to get:\[ 4 = 2x \]Divide both sides by 2 to solve for \( x \):\[ x = 2 \]Thus, the x-intercept is \((2, 0)\).

Step 2: Find the y-intercept

The y-intercept is the point where the line crosses the y-axis. This means the x-value is 0. To find the y-intercept, set \( x = 0 \) in the equation \( y = 2x - 4 \):\[ y = 2(0) - 4 \]Simplify to get:\[ y = -4 \]So, the y-intercept is \((0, -4)\).

Step 3: Plot the intercepts on a graph

On a coordinate plane, plot the points for the x-intercept \((2, 0)\) and the y-intercept \((0, -4)\). Each of these points represents where the line will cross the corresponding axis.

Step 4: Draw the line

Once the intercepts are plotted, use a ruler to draw a straight line through the points \((2, 0)\) and \((0, -4)\). This line represents the graph of the equation \( y = 2x - 4 \).

Key Concepts

Graphing

When it comes to graphing a linear equation like \( y = 2x - 4 \), the process is interesting and straightforward. The goal is to visually represent the equation on the coordinate plane. This involves identifying two key points called the intercepts. Once these points are located, you can draw a straight line through them. This line is the graph of the equation.
Graphing is a powerful tool because it translates an algebraic equation into a visual form. This helps you see the relationship between the variables. To start graphing, determine the intercepts by solving the equation with specific values for \( x \) and \( y \). Once identified, these intercepts can be used to form a line on the graph by connecting them.

X-intercept

The \( x \)-intercept is an essential concept in graphing linear equations. It represents the point where the line crosses the x-axis. At this point, the value of \( y \) is zero. Understanding how to find the \( x \)-intercept can help in plotting the graph of an equation.
To find the \( x \)-intercept of the equation \( y = 2x - 4 \), we set \( y = 0 \) and solve for \( x \):

• Replace \( y \) with 0: \( 0 = 2x - 4 \)
• Add 4 to both sides: \( 4 = 2x \)
• Divide both sides by 2: \( x = 2 \)

This results in the point \((2, 0)\), which is where the line crosses the x-axis. Recognizing this point enables placing part of the graph on the coordinate plane.

Y-intercept

The \( y \)-intercept is another crucial intercept in graphing. It is the point where the line crosses the y-axis, meaning the \( x \)-value at this point is zero. Finding the \( y \)-intercept is key to drawing the graph of a linear equation.
To find the \( y \)-intercept from the equation \( y = 2x - 4 \), set \( x = 0 \) and solve for \( y \):

• Replace \( x \) with 0: \( y = 2(0) - 4 \)
• This simplifies to \( y = -4 \)

Thus, the \( y \)-intercept is \((0, -4)\). This point marks where the line intersects with the y-axis.

Coordinate Plane

The coordinate plane is a critical part of understanding how graphs work. It's a two-dimensional surface allowing you to plot points, lines, and curves, like our line from the equation \( y = 2x - 4 \). It consists of two perpendicular axes: the x-axis, which runs horizontally, and the y-axis, which runs vertically. A point on this plane comprises an \( x \)-value and a \( y \)-value, written as \((x, y)\).
To graph a linear equation, use the coordinate plane to plot the intercepts you found. For our example, we plot the points \((2, 0)\) and \((0, -4)\):

• Plot each point where the line intersects the axes.
• Draw a line connecting these points using a ruler for precision.

This line visually represents the equation, showing the relationship between \( x \) and \( y \) as defined in the equation.