4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Some hills are hills, and some hills are mountains. Of course, some hills are alive with the sound of music, but that's a different story.
The difference between an easy and difficult ski slope isn't just height; a reasonably competent skier can handle a long, gentle slope. It's those steep ones that tend to be challenging and…entertaining. What makes a hill steep is its rate of change, the change in height for every change in length.
Students should know that to determine the rate of change, they should choose two points and subtract the differences in the height. Divide that number by the differences in horizontal location for those same two points. If this sounds eerily familiar to rise over run to your students, they're eerily perceptive. Slope is the rate of change.
Given a linear function or two points, students should be able to find the rate of change and the initial value (the y-intercept) and interpret it in terms of its context. Remembering that for rate of change, y values go on the numerator and x values go on the denominator (or the far easier "rise over run"). This is essential so that students don't calculate reciprocal slopes.
It's beneficial for students to apply slopes to functions, particularly linear functions. Knowing that the rate of change is constant for a linear function and changes for a nonlinear function is also important. They should know that a positive slope means uphill (the function is increasing) and a negative slope means downhill (the function is decreasing).
- Word Problems and One Line - Math Shack
- Writing Equations of Lines - Math Shack
- Matching Lines with Equations - Math Shack
- Linear Functions as Models - Math Shack
- Slope Point Word Problem - Math Shack
- Slope and Intercept - Math Shack
- Finding the Equations of Lines in Standard Form - Math Shack
- Point-Slope Form - Math Shack
- Graphing Lines Given Slope and Y-Intecept - Math Shack