Problem 2
Question
Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(2,1)\( and \)(3,4)$$
Step-by-Step Solution
Verified Answer
The slope of the line passing through the points (2,1) and (3,4) is 3 and the line rises.
1Step 1: Identify the coordinates.
The two points given are (2,1) and (3,4). So, (x1,y1) = (2,1) and (x2,y2) = (3,4).
2Step 2: Plug into the slope formula.
Substitute the coordinates into the slope formula. So, m = (4-1) / (3-2).
3Step 3: Calculate the slope.
Solve the expression obtained in the previous step. So, m = 3.
4Step 4: Interpret the slope.
The value of the slope is 3, which is a positive number. Therefore, the line rises.
Key Concepts
CoordinatesSlope FormulaPositive SlopeLine Rising
Coordinates
Coordinates help us identify specific points on a graph or a plane. They are expressed as ordered pairs, like \((x_1, y_1)\) and \((x_2, y_2)\). Each ordered pair consists of an x-value and a y-value, which corresponds to a point on the Cartesian plane.
- **First point**: \((2,1)\) means x-coordinate is 2 and y-coordinate is 1.- **Second point**: \((3,4)\) means x-coordinate is 3 and y-coordinate is 4.When we speak about coordinates, we mean the position of a point defined within a plane, allowing us to determine how points are located in relation to one another.
- **First point**: \((2,1)\) means x-coordinate is 2 and y-coordinate is 1.- **Second point**: \((3,4)\) means x-coordinate is 3 and y-coordinate is 4.When we speak about coordinates, we mean the position of a point defined within a plane, allowing us to determine how points are located in relation to one another.
Slope Formula
The slope of a line is a measure of how steep the line is or how much it "tilts" in a graph. It is calculated using the slope formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\] This formula helps us find the slope of the line that passes through two given points. Let's break it down:
In our exercise, plugging in the coordinates gives:
\[m = \frac{4 - 1}{3 - 2} = 3\]
\[m = \frac{y_2 - y_1}{x_2 - x_1}\] This formula helps us find the slope of the line that passes through two given points. Let's break it down:
- - **\(y_2 - y_1\)**: The difference between the y-coordinates of the given points.
- - **\(x_2 - x_1\)**: The difference between the x-coordinates of the given points.
In our exercise, plugging in the coordinates gives:
\[m = \frac{4 - 1}{3 - 2} = 3\]
Positive Slope
A positive slope indicates that as we move along the line from left to right, the line moves upwards. In mathematical terms, if \(m > 0\), then the slope is positive.
For instance, a slope of 3 means that for every unit increase in the x-direction, the y-value increases by 3 units.
This type of slope suggests that there's a positive relationship between the variables plotted on the x-axis and y-axis. The larger the positive number, the steeper the line.
For instance, a slope of 3 means that for every unit increase in the x-direction, the y-value increases by 3 units.
This type of slope suggests that there's a positive relationship between the variables plotted on the x-axis and y-axis. The larger the positive number, the steeper the line.
Line Rising
When describing a line as 'rising,' we refer to the direction it moves as we trace it from the left to the right side of a graph. A line rises if its slope is positive, showing that it ascends rather than descends.
This kind of line indicates growth or an increase in values. For our example, with a slope of 3, the line through the points\((2,1)\) and \((3,4)\) indeed rises, confirming the upward trajectory visually depicted on a graph.
Remember, a rising line means progress or elevation in the values represented by the points.
This kind of line indicates growth or an increase in values. For our example, with a slope of 3, the line through the points\((2,1)\) and \((3,4)\) indeed rises, confirming the upward trajectory visually depicted on a graph.
Remember, a rising line means progress or elevation in the values represented by the points.
Other exercises in this chapter
Problem 1
Find the distance between each pair of points. If necessary, round answers to two decimals places. $$(2,3) \text { and }(14,8)$$
View solution Problem 2
Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{2}-1 $$
View solution Problem 2
Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=6 x \text { and } g(x)=\frac{x}
View solution Problem 2
If \(f(x)=3 x^{2}-2 x+1\) and \(g(x)=4 x-1,\) find: a. \((f+g)(x)\) b. \((f+g)(5)\)
View solution