Problem 2
Question
Exer. 1-6: Sketch the line through \(A\) and \(B\), and find its slope \(m\). $$ A(4,-1), \quad B(-6,-3) $$
Step-by-Step Solution
Verified Answer
The slope of the line through points \( A(4,-1) \) and \( B(-6,-3) \) is \( \frac{1}{5} \).
1Step 1: Identify the coordinates of the points
We begin by identifying the coordinates of the given points. The coordinates of point \( A \) are \( (4, -1) \) and the coordinates of point \( B \) are \( (-6, -3) \).
2Step 2: Use the slope formula
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the coordinates of points \( A \) and \( B \) into this formula gives: \[ m = \frac{-3 - (-1)}{-6 - 4} \]
3Step 3: Calculate the slope
Simplify the expression obtained in Step 2: \[ m = \frac{-3 + 1}{-6 - 4} = \frac{-2}{-10} = \frac{1}{5} \] Thus, the slope of the line connecting points \( A \) and \( B \) is \( \frac{1}{5} \).
4Step 4: Sketch the line
To sketch the line, plot points \( A(4, -1) \) and \( B(-6, -3) \) on a coordinate plane. After plotting, draw a straight line passing through both points. This line represents the line determined by points \( A \) and \( B \) with a slope of \( \frac{1}{5} \).
Key Concepts
Coordinate GeometryPlotting PointsLinear Equations
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. The coordinate plane is a two-dimensional surface formed by two axes: the x-axis (horizontal) and the y-axis (vertical).
These axes intersect at a point called the origin, with coordinates (0, 0).
In this exercise, the points A and B have specific coordinates: A(4, -1) and B(-6, -3). These are plotted using ordered pairs where the first value indicates the position on the x-axis, and the second indicates the position on the y-axis. This method allows for precise placement of points on the plane and is critical in forming lines, curves, and other geometric figures.
These axes intersect at a point called the origin, with coordinates (0, 0).
In this exercise, the points A and B have specific coordinates: A(4, -1) and B(-6, -3). These are plotted using ordered pairs where the first value indicates the position on the x-axis, and the second indicates the position on the y-axis. This method allows for precise placement of points on the plane and is critical in forming lines, curves, and other geometric figures.
- Understanding the coordinate axes: The x-axis is horizontal, and the y-axis is vertical.
- Coordinates: Each point on the plane is represented by an ordered pair (x, y).
Plotting Points
Plotting points on a coordinate grid is a foundational skill in coordinate geometry. It involves locating a point using its coordinates.
In our example, point A is at (4, -1) and point B is at (-6, -3). This means for point A, move 4 units along the x-axis and then 1 unit down the y-axis. For point B, move 6 units left along the x-axis and 3 units down the y-axis.
Once the points are plotted, you can draw a line to visualize the connection between them. This concept is powerful, as it allows you to visualize mathematical problems effectively.
In our example, point A is at (4, -1) and point B is at (-6, -3). This means for point A, move 4 units along the x-axis and then 1 unit down the y-axis. For point B, move 6 units left along the x-axis and 3 units down the y-axis.
Once the points are plotted, you can draw a line to visualize the connection between them. This concept is powerful, as it allows you to visualize mathematical problems effectively.
- Use a ruler for precision when sketching lines.
- Ensure the scale used on the axes is consistent to maintain accuracy.
- Check your plotted points by verifying their coordinates on the axes.
Linear Equations
The concept of linear equations comes into play when understanding the lines formed by plotting points. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
When we talk about the line through points A and B with a slope of \( \frac{1}{5} \), we can represent it with a linear equation. The general form of a linear equation is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. In our case, with a slope of \( \frac{1}{5} \), the equation can define the behavior of the line between points A and B.
Understanding the slope, \( m \), of a line is crucial. Here, it represents the change in y for a unit change in x. In simpler terms, for every step to the right along the line, the height of the line increases by a fraction defined by the slope.
When we talk about the line through points A and B with a slope of \( \frac{1}{5} \), we can represent it with a linear equation. The general form of a linear equation is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. In our case, with a slope of \( \frac{1}{5} \), the equation can define the behavior of the line between points A and B.
Understanding the slope, \( m \), of a line is crucial. Here, it represents the change in y for a unit change in x. In simpler terms, for every step to the right along the line, the height of the line increases by a fraction defined by the slope.
- Slope: Determines the steepness of the line.
- General Equation Form: \( y = mx + c \) is key in determining a line's formula.
Other exercises in this chapter
Problem 2
Exer. 1-2: Find (a) \((f+g)(3)\) (b) \((f-g)(3)\) (c) \((f g)(3)\) (d) \((f / g)(3)\) $$ f(x)=-x^{2}, \quad g(x)=2 x-1 $$
View solution Problem 2
If \(f(x)=-x^{3}-x^{2}+3\), find \(f(-3), f(0)\), and \(f(2)\).
View solution Problem 2
Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ y=3 x+2 $$
View solution Problem 2
Plot the points \(A(-3,1), B(3,1), C(-2,-3), D(0,3)\), and \(E(2,-3)\) on a coordinate plane. Draw the line segments \(A B\), \(B C, C D, D E\), and \(E A\).
View solution