Question

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. $$ y=2 x+8 $$

Short answer

The y-intercept is \((0, 8)\) and the x-intercept is \((-4, 0)\).

Step-by-step solution

- Identify the y-intercept

To find the y-intercept, set \(x=0\) and solve for \(y\).Substitute \(x\) in the equation:\[ y = 2(0) + 8 \]Thus, \(y = 8\). The y-intercept is \((0, 8)\).

- Identify the x-intercept

To find the x-intercept, set \(y=0\) and solve for \(x\).Substitute \(y\) in the equation:\[ 0 = 2x + 8 \]Solve for \(x\):\[ -8 = 2x \]\[ x = -4 \]The x-intercept is \((-4, 0)\).

- Create a table of values

Choose a few values of \(x\), substitute them into the equation to find the corresponding \(y\) values.For example:1. \(x = -2\):\[ y = 2(-2) + 8 = 4 \] \ Point: \((-2, 4)\)2. \(x = 2\):\[ y = 2(2) + 8 = 12 \] \ Point: \((2, 12)\)

- Plot the points on a graph

Plot the intercepts \((0, 8)\) and \((-4, 0)\), as well as the additional points \((-2, 4)\) and \((2, 12)\) on a coordinate plane.

- Draw the line

Draw a straight line through all plotted points. This line is the graph of the equation \(y=2x+8\). Be sure to label the intercepts on the graph.

Key concepts

y-intercept

The y-intercept is the point where the graph of a linear equation crosses the y-axis. It represents the value of y when x equals zero. To find the y-intercept, substitute x with 0 in the given equation and solve for y. For the equation \( y = 2x + 8 \), setting \( x = 0 \) gives us:

\( y = 2(0) + 8 = 8 \).

Thus, the y-intercept is (0, 8). When plotting the graph, this point will always land on the y-axis. Remember, every y-intercept has an x-coordinate of zero.

x-intercept

The x-intercept is where the graph crosses the x-axis. At this point, y is zero. To find the x-intercept, set \( y = 0 \) in the equation and solve for x. For our equation \( y = 2x + 8 \), substitute y with 0:

\( 0 = 2x + 8 \).

Solve for x:

\( -8 = 2x \)
\( x = -4 \).

Hence, the x-intercept is (-4, 0). This point will always be on the x-axis and have a y-coordinate of zero.

plotting points

Plotting points involves choosing values for x, calculating the corresponding y values, and marking these coordinates on the graph. This method helps in creating the shape of the graph. Using the equation \( y = 2x + 8 \), let's choose some x-values and find their respective y-values.

• For \( x = -2 \):
  \( y = 2(-2) + 8 = 4 \) -> Point (-2, 4)
• For \( x = 2 \):
  \( y = 2(2) + 8 = 12 \) -> Point (2, 12)

By plotting these points (-2, 4), (2, 12), along with intercepts, you can see the general direction of the line.

coordinate plane

The coordinate plane, or Cartesian plane, is a two-dimensional surface defined by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). These lines intersect at the origin, designated as (0, 0). When graphing linear equations:

• Each point on the plane is defined by an (x, y) pair.
• The x-coordinate tells you how far to move horizontally from the origin.
• The y-coordinate indicates how far to move vertically.

Using this system, plot each point calculated and draw a straight line through them to represent the equation. This visual method aids in better understanding the behavior of linear equations.