Definition of Slope-Intercept Form
The slope-intercept form of a linear equation is a specific way to express a linear function. It is written as:
- Formula: y = mx + c
Here:
- m represents the slope of the line, which indicates the rate of change of y with respect to x.
- c is the y-intercept, representing the point where the line crosses the y-axis (i.e., when x = 0, y = c).
This form is particularly useful because it directly provides two key pieces of information about the line: its slope and its y-intercept, which are crucial for graphing and understanding the characteristics of the line.
Example 1: Identifying Slope and Intercept
Consider the equation y = 2x + 3. Here:
- m (Slope) = 2
- c (Y-Intercept) = 3
Graphing Using Slope-Intercept Form
Graphing a line using its slope-intercept form involves identifying the y-intercept and then using the slope to find another point on the line. Once two points are identified, the line can be drawn by connecting them.
Step 1: Plotting the Y-Intercept
The y-intercept (c) is the point where the line crosses the y-axis. Start by locating this point on the graph.
Step 2: Using the Slope to Find Another Point
The slope (m) is the ratio of the vertical change to the horizontal change between any two points on the line. From the y-intercept, move according to the slope to find the next point.
Step 3: Drawing the Line
Connect the two points with a straight line, and extend it in both directions.
Example 2: Graphing a Line
Consider the equation y = 3x - 2.
- Start by plotting the y-intercept at (0, -2).
- The slope is 3, which can be written as 3/1, meaning for every 3 units up (positive y-direction), move 1 unit to the right (positive x-direction). From (-2), move up 3 units and 1 unit to the right to find the next point (1, 1).
- Draw a line through (0, -2) and (1, 1).
Here is the graph of the equation y = 3x - 2:
The line intersects the y-axis at y = -2 (the y-intercept) and has a slope of 3, meaning for every unit increase in x, y increases by 3 units.
Applications and Significance
The slope-intercept form is not merely a mathematical representation but a tool that aids in understanding the relationship between two variables, interpreting trends, and making predictions in various fields like economics, physics, and social sciences.
Example 3: Real-World Application
Consider a scenario where a taxi company charges a flat fee of £5 and an additional £2 per mile travelled. The total cost (y) can be represented as: y = 2x + 5 Here, x represents the miles travelled. The slope (2) indicates that the cost increases by £2 for each additional mile, and the y-intercept (5) represents the initial charge regardless of the distance travelled.
Exam-Style Questions
Example 4: Identifying and Graphing
Given the equation y = -4x + 6, identify the slope and y-intercept and sketch the graph of the line.
- Slope (m): -4
- Y-Intercept (c): 6
To graph, start by plotting the y-intercept at (0, 6). Since the slope is -4 (or -4/1), from point (0, 6), move down 4 units and 1 unit to the right to find the next point (1, 2). Connect these points to graph the line.
Here is the graph of the linear equation y = -4x + 6:
The line intersects the y-axis at y = 6 (the y-intercept) and has a slope of -4, meaning for every unit increase in x, y decreases by 4 units.
Example 5: Formulating Equations
A line passes through the points (2, 3) and (4, 7). Find the equation of the line in slope-intercept form.
First, find the slope using m = (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 2. Using the point-slope form y - y1 = m(x - x1) and substituting point (2, 3) and m = 2, we get y - 3 = 2(x - 2). Simplifying, we find the equation of the line as y = 2x - 1.
Additional Insights into Slope-Intercept Form
Understanding the Slope
The slope (m) in the slope-intercept form is a measure of the steepness of the line. A positive slope indicates that the line ascends from left to right, while a negative slope indicates a descent. The magnitude of the slope determines the steepness: a larger absolute value of m results in a steeper line. For instance, in the equation y = 3x + 1, the slope is 3, indicating a relatively steep ascent. Conversely, in y = -0.5x + 4, the slope is -0.5, indicating a gentle descent.
Interpreting the Y-Intercept
The y-intercept (c) provides insights into the initial or starting value of the dependent variable (y) when the independent variable (x) is zero. In financial contexts, for instance, the y-intercept could represent a base fee or initial investment. In scientific experiments, it might represent a baseline measurement before any variables are manipulated.
Graphical Transformations
Modifying the values of m and c in the equation y = mx + c results in various graphical transformations:
- Changing m alters the steepness and direction of the line.
- Changing c shifts the line up or down without altering its steepness.
Example 6: Graphical Transformation
Consider the equation y = x + 2 and its transformation to y = 2x + 2. The slope has changed from 1 to 2, resulting in a steeper line, while the y-intercept remains the same, ensuring the line still crosses the y-axis at (0, 2).
Here are the graphs of the equations y = x + 2 and y = 2x + 2:
- The blue line represents y = x + 2. It has a slope of 1 and a y-intercept of 2, meaning it crosses the y-axis at (0, 2).
- The orange line represents y = 2x + 2. It has a steeper slope of 2 but maintains the same y-intercept of 2.
As you mentioned, the transformation from the blue line to the orange line involves an increase in the slope while keeping the y-intercept constant, resulting in a steeper line that still crosses the y-axis at the same point (0, 2).
Example 7: Analyzing Graphs
Given the equation y = -x + 5, the negative slope indicates a downward direction, and the y-intercept at (0, 5) indicates the starting point. The line will descend from left to right, crossing through the point (0, 5).
FAQ
No, the slope-intercept form cannot represent vertical lines. Vertical lines have an undefined slope since they go straight up and down, and the change in x (run) is zero. In the slope formula, m = (change in y) / (change in x), if the change in x is zero, the slope becomes undefined. Vertical lines are typically represented by equations of the form x = k, where k is a constant indicating the x-coordinate of all points on the line.
The letter 'c' in the slope-intercept form y = mx + c represents the constant term or the y-coordinate where the line intersects the y-axis. The choice of 'c' is arbitrary, and it stands for "constant". It's essential to understand that different textbooks or regions might use different letters, such as 'b', to represent the y-intercept. However, the concept remains the same.
To determine the slope of a line from an equation not in the slope-intercept form, you can rearrange the equation to get it into the form y = mx + c. Once in this form, the coefficient of x will be the slope. For instance, if you have the equation 2x - 3y = 6, you can rearrange it to get y = (2/3)x - 2. From this, you can see that the slope m is 2/3.
The slope-intercept form, y = mx + c, is designed to clearly display the slope (m) and y-intercept (c) of a linear equation. This form is particularly useful for graphing and understanding the characteristics of a line. On the other hand, the standard form of a linear equation is Ax + By = C, where A, B, and C are constants. The standard form doesn't directly provide the slope or y-intercept, but it's useful for understanding the relationship between x and y and for finding the x and y intercepts.
The slope of a line determines its steepness and direction. A positive slope indicates that the line rises as it moves from left to right, while a negative slope means the line falls. The greater the absolute value of the slope, the steeper the line. If the slope is 0, the line is horizontal, and if the slope is undefined, the line is vertical. The angle or inclination of the line with respect to the positive x-axis can be determined using trigonometry, specifically the tangent function, where the tangent of the angle is equal to the slope.
Practice Questions
To find the equation of the line passing through points A(2, 3) and B(4, 7), we first need to determine the slope (m). The slope is found using the formula m = (y2 - y1) / (x2 - x1). Substituting the given points, m = (7 - 3) / (4 - 2) = 2. Now, using the slope and point A in the point-slope form equation y - y1 = m(x - x1), we get y - 3 = 2(x - 2). Simplifying, we find the equation of the line as y = 2x - 1.
The given equation is y = 50x + 80, which is in slope-intercept form y = mx + c. Here, m (slope) = 50 and c (y-intercept) = 80. The slope, m = 50, represents the rate of change of the cost with respect to the hours worked, meaning the plumber charges £50 per hour of work. The y-intercept, c = 80, represents the initial cost of hiring the plumber, meaning there is a flat fee of £80 regardless of the number of hours worked. So, hiring the plumber costs £80 plus an additional £50 for each hour of work.
