Math - 2019-20
8.16 - Slope and Intercepts
8.16 The student will
a) recognize and describe the graph of a linear function with a slope that is positive, negative, or zero;
b) identify the slope and y-intercept of a linear function given a table of values, a graph, or an equation in y = mx + b form;
c) determine the independent and dependent variable, given a practical situation modeled by a linear function;
d) graph a linear function given the equation in y = mx + b form;
e) make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs.
- I can find the correlation between different types of
vehicles and the number of tires they have, the amount of money put into a
vending machine and the kind of selection that comes out, and track the money I
save over a year to determine when I can buy a cell phone.
- I will
recognize that functions are mathematical representations of many input-output
UNDERSTANDING THE STANDARD
· A linear function is an equation in two variables whose graph is a straight line, a type of continuous function.
· A linear function represents a situation with a constant rate. For example, when driving at a rate of 35 mph, the distance increases as the time increases, but the rate of speed remains the same.
· Slope (m) represents the rate of change in a linear function or the “steepness” of the line. The slope of a line is a rate of change, a ratio describing the vertical change to the horizontal change.
slope = =
· A line is increasing if it rises from left to right. The slope is positive (i.e., m > 0).
· A line is decreasing if it falls from left to right. The slope is negative (i.e., m < 0).
· A horizontal line has zero slope (i.e., m = 0).
· A discussion about lines with undefined slope (vertical lines) should occur with students in grade eight mathematics to compare undefined slope to lines with a defined slope. Further exploration of this concept will occur in Algebra I.
· A linear function can be written in the form y = mx + b, where m represents the slope or rate of change in y compared to x, and b represents the y-intercept of the graph of the linear function. The y-intercept is the point at which the graph of the function intersects the y-axis and may be given as a single value, b, or as the location of a point (0, b).
Given the equation of the linear function y
= −3x +2, the slope is −3 or and the
y-intercept is 2 or (0, 2).
- Example: The table of values represents a linear function.
In the table, the point (0, 2) represents the y-intercept. The slope is determined by observing the change in each y-value compared to the corresponding change in the x-value.
slope = m = = = −3
· The slope, m, and y-intercept of a linear function can be determined given the graph of the function.
- Example: Given the graph of the linear function, determine the slope and y-intercept.
Given the graph of a linear function, the y-intercept is found by determining where the line intersects the y-axis. The y-intercept would be 2 or located at the point (0, 2). The slope can be found by determining the change in each y-value compared to the change in each x-value. Here, we could use slope triangles to help visualize this:
slope = m = = = −3
· Graphing a linear function given an equation can be addressed using different methods. One method involves determining a table of ordered pairs by substituting into the equation values for one variable and solving for the other variable, plotting the ordered pairs in the coordinate plane, and connecting the points to form a straight line. Another method involves using slope triangles to determine points on the line.
- Example: Graph the linear function whose equation is y = 5x − 1.
In order to graph the linear function, we can create a table of values by substituting arbitrary values for x to determining coordinating values for y:
The values can then be plotted as points on a graph.
Knowing the equation of a linear function written in y = mx + b provides information about the slope and y-intercept of the function. If the equation is y = 5x − 1, then the slope, m, of the line is 5 or and the y-intercept is −1 and can be located at the point (0, −1). We can graph the line by first plotting the y-intercept. We also know,
slope = m = =
Other points can be plotted on the graph using the relationship between the y and x values.
Slope triangles can be used to help locate the other points as shown in the graph below:
· A table of values can be used in conjunction with using slope triangles to verify the graph of a linear function. The y-intercept is located on the y-axis which is where the x-coordinate is 0. The change in each y-value compared to the corresponding x-value can be verified by the patterns in the table of values.
· The axes of a coordinate plane are generally labeled x and y; however, any letters may be used that are appropriate for the function.
· A function has values that represent the input (x) and values that represent the output (y). The independent variable is the input value.
· The dependent variable depends on the independent variable and is the output value.
· Below is a table of values for finding the approximate circumference of circles, C = pd, where the value of p is approximated as 3.14.
- The independent variable, or input, is the diameter of the circle. The values for the diameter make up the domain.
- The dependent variable, or output, is the circumference of the circle. The set of values for the circumference makes up the range.
· In a graph of a continuous function every point in the domain can be interpreted. Therefore, it is possible to connect the points on the graph with a continuous line because every point on the line answers the original question being asked.
· The context of a problem may determine whether it is appropriate for ordered pairs representing a linear relationship to be connected by a straight line. If the independent variable (x) represents a discrete quantity (e.g., number of people, number of tickets, etc.) then it is not appropriate to connect the ordered pairs with a straight line when graphing. If the independent variable (x) represents a continuous quantity (e.g., amount of time, temperature, etc.), then it is appropriate to connect the ordered pairs with a straight line when graphing.
- Example: The function y = 7x represents the cost in dollars (y) for x tickets to an event. The domain of this function would be discrete and would be represented by discrete points on a graph. Not all values for x could be represented and connecting the points would not be appropriate.
- Example: The function y = −2.5x + 20 represents the number of gallons of water (y) remaining in a 20-gallon tank being drained for x number of minutes. The domain in this function would be continuous. There would be an x-value representing any point in time until the tank is drained so connecting the points to form a straight line would be appropriate (Note: the context of the problem limits the values that x can represent to positive values, since time cannot be negative.).
· Functions can be represented as ordered pairs, tables, graphs, equations, physical models, or in words. Any given relationship can be represented using multiple representations.
· The equation y = mx + b defines a linear function whose graph (solution) is a straight line. The equation of a linear function can be determined given the slope, m, and the y-intercept, b. Verbal descriptions of practical situations that can be modeled by a linear function can also be represented using an equation.
- Example: Write the equation of a linear function whose slope is and y-intercept is −4, or located at the point (0, −4).
The equation of this line can be found by substituting the values for the slope, m = , and the y-intercept, b = −4, into the general form of a linear function y = mx + b. Thus, the equation would be y = x – 4.
- Example: John charges a $30 flat fee to trouble shoot a personal watercraft that is not working properly and $50 per hour needed for any repairs. Write a linear function that represents the total cost, y of a personal watercraft repair, based on the number of hours, x, needed to repair it. Assume that there is no additional charge for parts.
In this practical situation, the y-intercept, b, would be $30, to represent the initial flat fee to trouble shoot the watercraft. The slope, m, would be $50, since that would represent the rate per hour. The equation to represent this situation would be y = 50x + 30.
· A proportional relationship between two variables can be represented by a linear function y = mx that passes through the point (0, 0) and thus has a y-intercept of 0. The variable y results from x being multiplied by m, the rate of change or slope.
· The linear function y = x + b represents a linear function that is a non-proportional additive relationship. The variable y results from the value b being added to x. In this linear relationship, there is a y-intercept of b, and the constant rate of change or slope would be 1. In a linear function with a slope other than 1, there is a coefficient in front of the x term, which represents the constant rate of change, or slope.· Proportional relationships and additive relationships between two quantities are special cases of linear functions that are discussed in grade seven mathematics.
- What is the
relationship among tables, graphs, words, and rules in modeling a given
Any given relationship can be represented by all four.
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
and describe a line with a slope
that is positive, negative, or zero (0).
· 8.16b1 Given a table of values for a linear function, identify the slope and y-intercept. The table will include the coordinate of the y-intercept.
· 8.16b2 Given a linear function in the form y = mx + b, identify the slope and y-intercept.
· 8.16b3 Given the graph of a linear function, identify the slope and y-intercept. The value of the y-intercept will be limited to integers. The coordinates of the ordered pairs shown in the graph will be limited to integers.
· 8.16c1 Identify the dependent and independent variable, given a practical situation modeled by a linear function.
· 8.16d1 Given the equation of a linear function in the form y = mx + b, graph the function. The value of the y-intercept will be limited to integers.
· 8.16e1 Write the equation of a linear function in the form y = mx + b given values for the slope, m, and the y-intercept or given a practical situation in which the slope, m, and y-intercept are described verbally.
· 8.16e2 Make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs.