Another representation commonly used for linear relationships is an equation, or symbolic representation.

Source: Enchanted Rock Panorama 2012, jujutacular, Wikimedia Commons

Tami and her family went hiking in the Enchanted Rock State Natural Area near Llano, Texas. The base of Enchanted Rock is 1,400 feet above sea level, and the top of Enchanted Rock is 425 feet above the ground. Enchanted Rock is one of the largest batholiths, or underground rock formations that have been uncovered by surface level erosion, in the United States.

Tami used the following equation to describe her family’s elevation above sea level, y, as a function of time since the hike began, x.

Tami and her family also enjoy hiking in other places. Match each equation with the description of the y-intercept by dragging the description to the space next to the equation. Hint: You may need to convert measurement units before comparing the equations. Use a calculator or an online unit conversion calculator to help you.

Pause and Reflect

How can you determine the changes in the y-intercepts from an equation in slope-intercept form, y = mx + b? Interactive popup. Assistance may be required.

Check Your Answer

The y-intercept is the point (0, b), so any changes relate to comparisons of the value of b between two or more equations.

Consider the situation: A hiker in Death Valley, California, begins her climb at 150 feet below sea level and uses the function y = 12.5x – 150 to predict her elevation, y, after she has been hiking for x minutes. How would the function change if her starting elevation were 30 feet higher? Interactive popup. Assistance may be required.

Check Your Answer

Her starting elevation would be −150 + 30 = −120 feet, so the new function would be y = 12.5x – 120.

Practice

1. Jonathan and Jessie were each driving from their separate homes to visit their grandparents. Jonathan used the equation y = 120 − 65x, where x represents the time (in hours) since he left his home, to determine the number of miles, y, remaining until he reached his grandparents’ home. Jessie used the equation y = 180 – 65x to determine the number of miles remaining until he reached his grandparents’ home. Which boy started out farther away from their grandparents’ home?

   Interactive popup. Assistance may be required.

   Need a hint?

   Compare the y-intercepts of both equations.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Jessie, who started out 180 miles from his grandparents’ home, started out farther away. Jonathan started out 120 miles from his grandparents’ home.
2. Sharon borrowed money to purchase a used car. She can use the equation y = 4,000 – 250x, where x represents the number of months she will be paying the loan, to determine y, the remaining loan balance. If Sharon has to borrow an additional $500 to purchase the car of her choice, what will be the new equation she can use?

   Interactive popup. Assistance may be required.

   Need a hint?

   Will the $500 affect the 4,000 term or the coefficient of 250?
   Interactive popup. Assistance may be required.

   Check Your Answer

   y = 4,500 – 250x
3. Jaime wants to purchase a gym membership with an activation fee of $65 and a monthly fee of $20. He can use the equation y = 20x + 65, where x represents the number of months that he has been a member of the gym, to determine y, the total cost of the gym membership. If the gym has a special that reduces the activation fee by 25%, what will be the new equation that Jaime can use?

   Interactive popup. Assistance may be required.

   Need a hint?

   If the activation fee is reduced by 25%, what portion of the $65 will Jaime need to pay?
   Interactive popup. Assistance may be required.

   Check Your Answer

   y = 20x + 48.75