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Intercepts and the Cover-Up Method

Graph functions in standard form using intercepts

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Intercepts and the Cover-Up Method

Suppose you had a cat, and you gave him 1 treat per day. What if you wanted to graph the number of treats left in his treat jar as a function of the number of days that have passed? You could do it if you knew the intercepts, but how would you find them? 

Intercepts and the Cover-Up Method

Recall that to find the \begin{align*}x-\end{align*}intercept, substitute zero for the \begin{align*}y\end{align*}-value and that to find the \begin{align*}y-\end{align*}intercept, substitute zero for the \begin{align*}x\end{align*}-value. This method works for any form of linear equations.

A second method of finding the intercepts is called the Cover-Up Method. Using the Multiplication Property of Zero \begin{align*}a(0)=0\end{align*}, you can “cover-up” the other variable and solve for the intercept you wish to find. This method only works for linear equations that are in the form \begin{align*}ax+by=c.\end{align*} 

The cover-up method simply removes one variable at a time, in order to solve for the other variable. The solutions are then the intercepts, since removing a variable, or "covering up" a variable, is the same thing as setting it equal to zero. 

Let's use the Cover-Up Method to complete the following problems:

  1. Graph \begin{align*}-7x-3y=21\end{align*}.

To solve for the \begin{align*}y-\end{align*}intercept we set \begin{align*}x=0\end{align*} and cover up the \begin{align*}x\end{align*} term:

\begin{align*}-3y& =21\\ y& =-7 && (0,-7) \ \text{is the} \ y-intercept \text{.}\end{align*}

To solve for the \begin{align*}x-\end{align*}intercept, cover up the \begin{align*}y-\end{align*}variable and solve for \begin{align*}x\end{align*}:

\begin{align*}-7x& =21\\ x& =-3 && (-3,0) \ \text{is the} \ x-intercept \text{.}\end{align*}

Now graph by first plotting the intercepts and then drawing a line through these points.

  1. Jose has $30 to spend on food for a class barbecue. Hot dogs cost $0.75 each (including the bun) and burgers cost $1.25 (including bun and salad). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, spending exactly $30.

Begin by translating this sentence into an algebraic equation. Let \begin{align*}y=\end{align*} the number of hot dogs and \begin{align*}x=\end{align*} the number of burgers.

\begin{align*}1.25(x)+ 0.75(y)=30\end{align*}

Find the intercepts of the graph. This example will use the Cover-Up Method. Feel free to use substitution if you prefer.

\begin{align*}0.75y& =30\\ y& =40 && y-intercept(0,40)\end{align*}

\begin{align*}1.25x& =30\\ x& =24 && x-intercept(24,0)\end{align*}

By graphing Jose’s situation, you can determine the combinations of hot dogs and burgers he can purchase for exactly $30.00.

  1. Graph \begin{align*}y+\frac{3}{2}x=-3\end{align*} using the Cover-Up Method.

First start by "covering up" \begin{align*}y\end{align*}. This results in:

\begin{align*}\text{Start with the equation.} && y+\frac{3}{2}x&=-3\\ \text{Then``cover-up" y.} && \frac{3}{2}x&=-3\\ \text{Solve for x.} && x&=-3 \cdot \frac{2}{3}= -2 \end{align*}

When \begin{align*}y=0\end{align*}, \begin{align*}x=-2\end{align*}. This is the \begin{align*}x\end{align*}-intercept.

Now, "cover-up" \begin{align*}x\end{align*}:

\begin{align*}\text{Start with the equation.} && y+\frac{3}{2}x&=-3\\ \text{Then ``cover-up" x.} && y&=-3\\ \end{align*}

When \begin{align*}x=0\end{align*}, \begin{align*}y=-3\end{align*}. This is the \begin{align*}y\end{align*}-intercept.

Now graph the equation by plotting the two intercepts and connecting them with a line.

Examples

Example 1

Earlier, you were told to suppose that you had a cat that you gave 1 treat to per day and you wanted to graph the number of treats left in his treat jar as a function of the number of days that have passed. How would you find the intercepts?

You could use substitution or the Cover-Up Method. To use the Cover-Up Method, first put the equation into \begin{align*}ax+by=c\end{align*} form. Then, "cover up"  \begin{align*}y\end{align*} to find the \begin{align*}x\end{align*}-intercept. Finally, "cover up" \begin{align*}x\end{align*} to find the \begin{align*}y\end{align*}-intercept. With those, you can easily graph the equation.

Example 2

Graph \begin{align*}4y-3x=12\end{align*} using the cover-up method.

First start by "covering up" \begin{align*}y\end{align*}. This results in:

\begin{align*}\text{Start with the equation.} && 4y-3x&=12\\ \text{Then ``cover-up" y.} &&-3x&=12\\ \text{Solve for x.} && x&=-4 \end{align*}

When \begin{align*}y=0\end{align*}, \begin{align*}x=-4\end{align*}. This is the \begin{align*}x\end{align*}-intercept.

Now, "cover-up" \begin{align*}x\end{align*}:

\begin{align*}\text{Start with the equation.} && 4y-3x&=12\\ \text{Then ``cover-up" x.} && 4y&=12\\ \text{Solve for y.} && y&=3 \end{align*}

When \begin{align*}x=0\end{align*}, \begin{align*}y=3\end{align*}. This is the \begin{align*}y\end{align*}-intercept.

Now graph the equation by plotting the two intercepts and connecting them with a line.

Review

  1. Explain the process of the Cover-Up Method.
  2. Do you prefer the Substitution Method or the Cover-Up Method? Why?

Find the intercepts of the following equations using the Cover-Up Method.

  1. \begin{align*}5x-6y=15\end{align*}
  2. \begin{align*}3x-4y=-5\end{align*}
  3. \begin{align*}2x+7y=-11\end{align*}
  4. \begin{align*}5x+10y=25\end{align*}

In 7 – 18, use any method to find the intercepts and then graph the equation.

  1. \begin{align*}y=2x+3\end{align*}
  2. \begin{align*}6(x-1)=2(y+3)\end{align*}
  3. \begin{align*}x-y=5\end{align*}
  4. \begin{align*}x+y=8\end{align*}
  5. \begin{align*}4x+9y=0\end{align*}
  6. \begin{align*}\frac{1}{2} x+4y=12\end{align*}
  7. \begin{align*}x-2y=4\end{align*}
  8. \begin{align*}7x-5y=10\end{align*}
  9. \begin{align*}4x-y=-3\end{align*}
  10. \begin{align*}x-y=0\end{align*}
  11. \begin{align*}5x+y=5\end{align*}
  12. \begin{align*}7x-2y=-6\end{align*}
  13. What needs to be done to the following equation before you can use either method to find its intercepts? \begin{align*}3(x+2)=2(y+3)\end{align*}
  14. At the local grocery store, strawberries cost $3.00 per pound and bananas cost $1.00 per pound. If I have $10 to spend between strawberries and bananas, draw a graph to show what combinations of each I can buy and spend exactly $10.
  15. A movie theater charges $7.50 for adult tickets and $4.50 for children. If the $900 theater takes in ticket sales for a particular screening, draw a graph that depicts the possibilities for the number of adult tickets and the number of child tickets sold.
  16. In football, touchdowns are worth 6 points, field goals are worth 3 points, and safeties are worth 2 points. Suppose there were no safeties and the team scored 36 points. Graph the situation to determine the combinations of field goals and touchdowns the team could have had.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.6. 

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Vocabulary

intercept

The point at which a graphed equation crosses an axis. The x-intercept is an ordered pair at which the line crosses the x-axis (the horizontal axis). Its ordered pair has the form (x,0). The y-intercept is an ordered pair at which the line crosses the y-axis (the vertical axis). Its ordered pair has the form (0,y)

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