NASA is planning a special project for Valentine's Day--Mission4Love. They want to show a live sunset from Mars on national television.
In order to get the best view, they plan to send the Mars Rover to the top of a mountain. To complete the mission successfully, NASA has to figure out if the Rover has enough power to reach the top of the mountain and which route he has to take. We can help NASA by using slope-intercept form.
Introduction to Slope-Intercept Form & Formula
There are three common forms to write linear equations:
- Slope-Intercept form
- Point-Slope form
- Standard form
Today we will focus on Slope-Intercept form. As you can see, the slope is explicitly included in this form. Let's recall what slope is:
- Slope shows the steepness of a line.
- It can be calculated by "rise over run".
- If you take any two points on the line, you can calculate the slope using this formula: m equals the change of the two y-coordinates divided by the change of the two x-coordinates, or delta y over delta x for short. The Greek letter Delta means 'change' in math expressions.
So, what about about the intercepts? Let's look at the formula for the Slope-Intercept form. Remember, M represents the slope of the line, while b is the y-intercept.
Y-Intercept
Lets look at the graph of y=2x+1 in the coordinate plane.
The y-intercept is the point where the line crosses the y-axis. Here, the line crosses the y-axis at positive 1.
What happens if you change the y-intercept to 2? As you can see, the slope stays the same, but the whole line is shifted up one unit.
What happens if you change the y-intercept to -1? As you can see, the slope stays the same, but the whole line is shifted down by two units.
Slope
Now that we understand the concept of the y-intercept, let's look at slope.
Here, the slope is positive 2, meaning the graph is increasing from left to right. For every unit you go to the right, you must go up 2 units.
If we let m equal 1, the y-intercept stays the same, but now the slope is different. It still increases from left to right, but for every 1 unit you go to the right, you must also go up 1 unit.
If you change the slope to be zero, the y-intercept stays the same, but the line becomes horizontal. It is neither increasing nor decreasing.
Let's try changing the slope to −1. Again, the y-intercept stays the same, but the line is now decreasing from left to right because it has a negative slope. Now, for every unit right, you must go down one unit. Now that you've gotten the hang of slope-intercept form, let's get back to our Mars Rover.
Slope-Intercept Form Example 1
The Rover currently has 20 units of solar battery power left. If it uses 2 units of battery power per mile, how many miles can the Rover drive before it runs out of battery? Can it reach a spot that is 8 miles away?
Let's put this information into slope-intercept form:
- x represents the number of miles traveled by the rover.
- y represents the amount of battery power left. We need to find the numbers to replace m and b.
- The battery power decreases by 2 units each mile, so you should put −2 as the slope.
- Since the Rover starts with 20 units of battery power, 20 is the y-intercept.
So how many miles can the Rover travel until the battery power gets down to 0 units? We put 0 in for y, the amount of battery power left. Now you can solve this equation to find x, the number of miles traveled:
- First, add 2x to both sides of the equation.
- Now you can divide both sides by 2.
And x=10. So, this means that the rover can travel 10 more miles with it's power. It can easily reach the spot that is 8 miles away.
If you graph y=−2x+20, you can see the solution right away: the x-intercept is 10. This is the point where now power is left.
Slope-Intercept Form Example 2
Let's look at one more problem. The Rover has to get to the top of the mountain to F (7,6). It can either take a route starting here at S1 (1,0) or here at S2 (5,0). The rover can go a maximum slope of 2.5. Which route is better?
Since the Rover can't go over anything with a slope higher than 2.5, we need to find the slope of two different routes. First let's find the slope of S2(5,0) to F(7,6).
The slope formula is y2−y1 over x2−x1. So, the first slope is 6−0 over 7−5. This reduces to 6 over 2 which is 3. That slope is too steep! Hopefully the other starting point will be better.
The second slope from S1(1,0) to F(7,6). Is 6−0 over 7−1. This reduces to 6 over 6 which is 1. This slope is better for the Rover.
It looks like we are pretty much done. But unfortunately, the Rover needs the whole formula to get into perfect position for the sunset. To find the y-intercept, we can plug-in all of the other information we know.
The slope, m, is 1. Now we can choose a point (x,y) on the route to plug in values for x and y. Let's choose the final point (7,6). 6 goes in for y. 7 goes in for x.
Now we can solve for b. Subtract 7 from both sides. B equals negative 1. So the equation for the Rover's route is y=x−1.
NASA managed to get the Rover in the perfect position for a romantic sunset. But what is this? Love is in the air on planet Mars.
