Slope

Slope exercise

• Label the picture using the correct terms.

  Hints

  For the line shown on the graph, the change in height and the change in width can be expressed in $3$ different ways.

  If $\Delta y$ increases and $\Delta x$ stays the same, $m$ also increases.

  Remember, slope is rise over run.

  Solution

  We use the following formula to calculate the slope of a line:

  $\text{slope}=\frac{\text{change in height}}{\text{change in width}}$.

  Mathematically, we write this formula in the following manner:

  $m=\frac{\Delta y}{\Delta x}$.

  • $m$ stands for the slope
  • $\Delta$ is the Greek capital letter delta, meaning 'change'
  • $\Delta y$ or $\Delta x$ is the change in $y$ or $x$, respectively
  • We can also expand $\Delta y$ to $y_2-y_1$, which is the difference between two y-coordinates. Similarly, $\Delta x$ can also be expanded to $x_2-x_1$, which is the difference between two x-coordinates
  • Another way to think about $\Delta y$ is rise and another way to think about $\Delta x$ is run
  • The pair of points, (x_1, y_1) and (x_2, y_2) is your choice. It also doesn't matter which is your "first" point and which is your "second" point; you just have to be consistent.

  For example: Point 1: ($x_1$, $y_1$) = (2, 20) Point 2: ($x_2$, $y_2$) = (10, 25)

  Point 1 "FIRST"

  $\large m = \frac{20-25}{2-10} = \frac{-5}{-8} = \frac{5}{8}$

  Point 2 "FIRST"

  $\large m = \frac{25-20}{10-2} = \frac{5}{8}$

• Determine the slope of the mountainside.

  Hints

  Pay attention to the order of subtraction:

  It doesn't matter which point you choose as point $1$ or point $2$, but when finding the slope of a line, the order in which you subtract the numbers must be consistent.

  The first coordinate of a point is the x-coordinate, and the second is the y-coordinate.

  Remember, another way to remember slope is rise over run.

  Reducing a fraction to its lowest terms means dividing the numerator and the denominator by the GCF.

  Solution

  We use the formula $m=\frac{\Delta y}{\Delta x}$ to evaluate the slope, $m$.

  • $\Delta y$ means change in height (or rise)
  • $\Delta x$ means change in width (or run)

  Let's find the slope using two points on the line: $(6,3)$ and $(15,6)$:

  $m=\frac{6-3}{15-6}=\frac39=\frac13$

  It's very important to pay attention to the order of subtraction. It doesn't matter which point you choose as point $1$ or point $2$, but when finding the slope of a line, the order in which you subtract the numbers has to be consistent.

• Examine the slopes of the lines using the formula $m = \frac{\Delta y }{\Delta x}$.

  Hints

  You need to know two points on a line to calculate the slope.

  For example $(0.5,2.5)$ and $(2.5,3)$ are two points on the violet line.

  Lines that are parallel to the x-axis have a slope of zero because the y-coordinates of the two points are the same.

  A rising line has a positive slope.

  Solution

  To use the slope formula, we need two points from each line.

  First, let's take a look at the blue line:

  • The points are $(1,4.5)$ and $(3,0.5)$
  • $m=\frac{0.5-4.5}{3-1}=-\frac42=-2$

  Next, we examine the green line:

  • The points are $(0,3.5)$ and $(3,3.5)$
  • we should have a zero slope since the y-coordinates of both points are the same
  • $m=\frac{3.5-3.5}{3-0}=\frac03=0$

  Moving on to the red line:

  • The points are $(1,2)$ and $(4,5)$
  • $m=\frac{5-2}{4-1}=\frac33=1$

  Finally, we look at the violet line:

  • The points are $(0.5,2.5)$ and $(2.5,3)$
  • $m=\frac{3-2.5}{2.5-0.5}=\frac{0.5}2=\frac14=0.25$

• Determine the slopes of the different routes.

  Hints

  The larger the slope, the steeper the line.

  Use the following formula to help you:

  $\large{m=\frac{\Delta y}{\Delta x}}$

  We only have positive slopes. Poor yetis!

  Pay attention to the order of subtraction.

  For example, for the following points:

  $\large{(x_1,y_1)}$ and $\large{(x_2,y_2)}$

  Though it doesn't matter which point is point 1 and which is point 2, you do have to be consistent.

  $\large{m=\frac{y_2-y_1}{x_2-x_1}}$

  Solution

  For each route, we use the formula:

  $\large{m=\frac{\Delta y}{\Delta x}}$

  Let's take a look at the slopes for the different routes.

  Yellow Route:

  • $m=\frac{4-2}{5-1}=\frac24=\frac12$

  Red Route:

  • $m=\frac{5-3}{4-2}=\frac22=1$

  Blue Route:

  • $m=\frac{7-1}{4-1}=\frac63=2$

  Green Route:

  • $m=\frac{10-4}{3.5-2}=\frac6{1.5}=4$

• Describe how the slope influences the look of a line in the coordinate plane.

  Hints

  Have a look at two points lying on a line parallel to the x-axis, such as $(3,3)$ and $(5,3)$.

  Use the equation $\large{m=\frac{\Delta y}{\Delta x}}$ to determine the slope of a line that passes through these points.

  What's going on if you have a line parallel to the y-axis that passes through the points $(3,3)$ and $(3,5)$?

  Use the equation $\large{m=\frac{\Delta y}{\Delta x}}$ to determine the slope.

  Remember! Since you're not allowed to divide by zero, any slope that has zero in the denominator is undefined!

  Solution

  Take a look at the whole slope tree:

  • on the left the line rises, so the slope is positive
  • on the right the line falls, so the slope is negative

  Parallel to the ground, our x-axis, we have a line with slope of zero. All lines parallel to this line have the same slope.

  But what's going on with the tree trunk? A line parallel to the y-axis always indicates an undefined slope.

  Hint: You can always tell a line will have an undefined slope if the $x$ coordinates of the two points are the same.

• Calculate the height of the mountain.

  Hints

  The x-coordinate of the mountain peak is $15+20=35$.

  The corresponding y-coordinate is the unknown height.

  Use the formula $\large{m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}}$ and plug in all the known values.

  In this case, $(x_2,y_2)$ are the coordinates of the mountain peak. The unknown value is $y_2$.

  Isolate the unknown value by using opposite operations.

  Solution

  The given information for this problem is:

  • the slope ($m=2$)
  • point on the mountain side ($15, 7$)
  • x-coordinate of the mountain peak ($15 + 20, y_2$)

  What we are looking for is the y-coordinate of the mountain peak, or the height.

  We need to use the formula $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$ and plug in the known values:

  $2=\frac{y_2-7}{35-15}=\frac{y_2-7}{20}$

  Multiplying by $20$ we get the following equation:

  $40=y_2-7$

  Performing opposite operations, we add $7$ to both sides of the equation, giving us the solution:

  $y_2=47$

  The height of the mountain peak is 47 units tall.