Lesson 3: Equations and Graphs

New Concept

Today, we'll see how equations come to life as graphs! You'll learn to translate an algebraic equation into a visual picture, making it easier to understand the relationship between variables and even solve for unknown values.

What’s next

Next up, you'll build your first graph in an interactive example. Then, you'll tackle a series of practice cards to master reading and creating them.

Property

A graph has two axes, horizontal and vertical, and the values for the variables are displayed along the axes. The first, or input, variable is displayed on the horizontal axis. The second, or output, variable is displayed on the vertical axis.

Examples

• For the equation $y = x + 5$, the input variable $x$ is on the horizontal axis and the output variable $y$ is on the vertical axis.
• In the equation $P = 2s$, the input $s$ is on the horizontal axis and the output $P$ is on the vertical axis.
• If an equation relates distance $d$ and time $t$ as $d = 50t$, time $t$ is the input on the horizontal axis and distance $d$ is the output on the vertical axis.

Explanation

A graph is a picture showing how two variables are related. The input value (what you start with) goes on the horizontal axis (bottom), and the output value (the result) goes on the vertical axis (side).

Property

We write the coordinates of pointinside parentheses as an ordered pair. The order of the coordinates makes a difference. We always list the horizontal coordinate first, then the vertical coordinate.

Examples

• The ordered pair $(5, 8)$ represents a point that is 5 units along the horizontal axis and 8 units along the vertical axis.
• To plot the point $(2, 9)$, you start at the origin, move 2 units horizontally, and then 9 units vertically.
• The point with coordinates $(0, 6)$ is located on the vertical axis, 6 units up from the origin.

Explanation

An ordered pair is like an address for a point on a graph. The first number tells you how far to move along the horizontal axis, and the second number tells you how far to move up or down the vertical axis.

Property

An ordered pair that makes an equation true is called a solution of the equation. Each point on the graph represents a solution of the equation.

Examples

• The ordered pair $(4, 10)$ is a solution to the equation $y = x + 6$ because substituting the values gives $10 = 4 + 6$, which is true.
• For the equation $w = 3z$, the ordered pair $(5, 15)$ is a solution because $15 = 3 \times 5$.
• The point $(2, 8)$ is a solution to $y = 10 - x$ because when you substitute, you get the true statement $8 = 10 - 2$.

Explanation

A solution is an ordered pair that fits perfectly into an equation. When you substitute the values, the equation becomes a true statement. Every single point on a graph's line is a solution to its equation.

Property

To graph an equation.

1. Make a table of values.
2. Choose scales for the axes.
3. Plot the points and connect them with a smooth curve.

Examples

• To graph $y = x + 3$, you can make a table. If $x=1$, $y=4$, so you plot the point $(1, 4)$.
• For the equation $y = 5x$, choose a value for $x$, like $x=2$. The corresponding $y$ is $10$. This gives the point $(2, 10)$ to plot on your graph.
• To graph $y = 12 - x$, if you pick $x=5$, then $y=7$. You would then plot the ordered pair $(5, 7)$ as one point on your graph's line.

Explanation

Graphing an equation is like connecting the dots. First, you create a table of solutions (ordered pairs). Then, you plot these points on the graph and draw a line or curve through them to see the full picture.

Property

We can use the graph to answer two types of questions about an equation:

1. Given a value of the input variable, find the corresponding value of the output variable.
2. Given a value of the output variable, find the corresponding value of the input variable.

Examples

• Using a graph of $y=3x$, to find $y$ when $x=4$, locate 4 on the horizontal axis, move up to the line, and read the corresponding $y$-value, which is 12.
• Using a graph of $y=x+5$, to solve $x+5=11$, find 11 on the vertical axis, move across to the line, and read the corresponding $x$-value, which is 6.
• On the graph of $y=x-2$, to evaluate for $x=8$, you find 8 on the horizontal axis, go up to the graph, and see that the $y$-value is 6.

Explanation

A graph lets you solve problems visually. To find an output, start on the horizontal axis, move to the line, then across to the vertical axis. To find an input, start on the vertical axis and do the reverse.