Analytic Geometry — Slope from Two Points (Part B)
Analytic Geometry — Slope from Two Points (Part B). Practice questions to deepen understanding of computing slope from two points and understanding its meaning. Online math practice with full solutions and step-by-step explanations.
Computing slope from two points — slope formula, meaning of positive, negative, and zero slope, and identifying slope from a graph. Practice with detailed explanations.
Calculate the slope of the line passing through the points \((1,2)\) and \((3,6)\).
We use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Here: \(\frac{6 - 2}{3 - 1} = \frac{4}{2} = 2\).
Given the points \((4,1)\) and \((8,5)\). What is the slope of the line passing through them?
Either point can be chosen as the first — just keep the same order in the numerator and denominator.
\(\frac{5 - 1}{8 - 4} = \frac{4}{4} = 1\).
Calculate the slope of the line passing through \((1,5)\) and \((3,1)\).
\(\frac{1 - 5}{3 - 1} = \frac{-4}{2} = -2\), therefore the slope is negative and the line is decreasing.
What is the slope of the line passing through \((3,1)\) and \((1,5)\)?
Even if the order of the points is swapped, the slope is the same as long as the same order is maintained in the formula:
\(\frac{5 - 1}{1 - 3} = \frac{4}{-2} = -2\).
Find the slope of the line passing through \((1,3)\) and \((5,3)\).
Here \(y_2 - y_1 = 3 - 3 = 0\), therefore \( m = 0\) and the line is horizontal.
What is the slope of the line passing through \((2,1)\) and \((2,5)\)?
Here \(x_2 - x_1 = 2 - 2 = 0\), and division by 0 is undefined. This is a vertical line.
Calculate the slope of the line passing through \((0,1)\) and \((4,3)\).
\(\frac{3 - 1}{4 - 0} = \frac{2}{4} = \frac{1}{2}\).
Find the slope of the line passing through \((2,4)\) and \((6,2)\).
\(\frac{2 - 4}{6 - 2} = \frac{-2}{4} = -\frac{1}{2}\).
Calculate the slope of the line passing through \((-2,3)\) and \((4,6)\).
\(\frac{6 - 3}{4 - (-2)} = \frac{3}{6} = \frac{1}{2}\).
Find the slope of the line passing through \((-3,-1)\) and \((1,1)\).
\(\frac{1 - (-1)}{1 - (-3)} = \frac{2}{4} = \frac{1}{2}\).
The line passes through \((1,2)\) and \((4,-1)\). Is the slope positive, negative, or zero?
As x increases from 1 to 4, the value of y decreases from 2 to (-1), therefore the slope is negative.
The line passes through \((0,-2)\) and \((3,4)\). What is the slope?
\(\frac{4 - (-2)}{3 - 0} = \frac{6}{3} = 2\).
The following table shows x and y values of a line: x 1 2 3 y 4 6 8 What is the slope?
When x increases by 1, y increases by 2. Therefore \(m = \frac{\Delta y}{\Delta x} = \frac{2}{1} = 2\).
Table: x 0 1 2 y 5 3 1 What is the slope?
For every increase of 1 in x, y decreases by 2. Therefore the slope is \(-2\).
In the following graph, two points A and B are marked on a line. Calculate the slope using the coordinates. A\((1,1)\), B\((4,3)\)
\(\frac{3 - 1}{4 - 1} = \frac{2}{3}\). The student uses the written coordinates, not necessarily reading from the diagram.
Given points A\((2,1)\) and B\((5,4)\) on a line. What is the slope?
\(\frac{4 - 1}{5 - 2} = \frac{3}{3} = 1\). The vectors only illustrate the concept \(\Delta x, \Delta y\).
Two lines have the following equations: \(y = 2x + 1\) \(y = 2x - 3\) What can be said about their slopes?
In both equations the slope is 2, therefore the lines are parallel.
Line a has slope \(m_1 = 1\) and line b has slope \(m_2 = -2\). Which is steeper?
We check the slope by absolute value: \(|-2| > |1|\), so line b is steeper (even though it is decreasing).
Two lines are perpendicular to each other. The first line has slope \(m_1 = 2\). What could the slope of the second line be?
For perpendicular lines: \(m_1 \cdot m_2 = -1\), therefore here \(m_2 = -\frac{1}{2}\).
In a certain line, as \(x\) increases, the value of \(y\) always remains 4. What is the slope?
If y does not change, there is no "rise" as x increases → slope is zero.
In another line, the value of \(x\) is always equal to 3, regardless of y. What can be said about the slope?
An equation of the form \(x = c\) describes a vertical line with no defined slope.
In a certain line segment, when x increases by 2, the value of y increases by 6. What is the slope?
The slope is \(\frac{\Delta y}{\Delta x} = \frac{6}{2} = 3\).
In another line, for every increase of 1 in x, y decreases by 3. What is the slope?
A decrease of 3 for every increase of 1 → slope \(-3\).
Which line has greater steepness? Line a with slope \(m = \frac{1}{2}\) Line b with slope \(m = 3\)
The larger the slope (in absolute value), the steeper the line. Here 3 > 0.5.
Line a has points \((0,0)\), \((2,4)\). Line b has points \((0,0)\), \((2,2)\). Which is steeper?
Slope of a: \(\frac{4-0}{2-0} = 2\). Slope of b: \(\frac{2-0}{2-0} = 1\). 2 > 1 → line a is steeper.
Line c has points \((1,2)\), \((3,6)\). Line d has points \((0,0)\), \((2,4)\). What is the relationship between them?
In both cases the slope is \(m = 2\), therefore the lines are parallel.
Two points are marked on a line: A\((1,4)\), B\((5,6)\). What is the slope?
\(\frac{6 - 4}{5 - 1} = \frac{2}{4} = \frac{1}{2}\).
Calculate the slope of the line passing through \((2,-3)\) and \((10,1)\).
\(\frac{1 - (-3)}{10 - 2} = \frac{4}{8} = \frac{1}{2}\).
Line a has slope \(m = \frac{3}{2}\) and line b has slope \(m = \frac{1}{3}\). What is true?
\(\frac{3}{2} > \frac{1}{3}\) therefore line a is steeper.
In a graph of distance (y) as a function of time (x), the slope of the line is \(m = 60\). What does this mean?
The slope represents the change in y for every change of 1 in x. Here: 60 distance units per unit of time.
Line a: \((0,0)\), \((1,4)\). Line b: \((0,0)\), \((1,2)\). Which is steeper?
Slope of a is 4, slope of b is 2, therefore line a is steeper.
It is known that on a certain line, when x increases from 2 to 6, y increases from 1 to 9. What is the slope?
\(\frac{9 - 1}{6 - 2} = \frac{8}{4} = 2\).
The line passes through \((0,0)\) and \((6,1)\). What is the slope?
\(\frac{1 - 0}{6 - 0} = \frac{1}{6}\).
The line passes through \((2,5)\) and \((7,5)\). What is the slope and what can be said about the line?
When y does not change, the slope is \(0\) and the line is horizontal.
The line passes through \((4,1)\) and \((4,-3)\). What can be said about the slope?
Here \(x\) is constant, therefore this is a vertical line with no defined slope.