Modeling with Functions
What is the standard form of a linear equation?
Ax+By=C
What are two common types of functions used to model real-world data?
Linear and exponential
What is a system of equations?
A system of equations is a set of two or more equations that contain the same variables. The goal is to find the values for the variables that make all of the equations in the system true at the same time.
What are the four types of transformations in geometry?
Translation (slides), rotation (turn), reflection (flip), and dilation (shrink/grow)
How do you identify the slope from a linear equation in slope-intercept form?
y=mx+b
m = slope, b = intercept, x and y are variables
How can you determine which model is a better fit for a set of data?
Graphing; see which shape aligns with the majority of the points plotted
Describe how to solve a system of equations using the substitution method.
1. Isolate the variable
2. Substitute the expression
3. Solve the new equation
4. Find the second variable
5. Write the solution
6. Check your answer
How does a translation affect the coordinates of a shape?
(x,y) → (x+a,y+b)
Add or subtract a value to the original points to get the new translated point.
Write an equation in slope intercept form if:
m = -3/2 and b = 5
y = (-3/2)x + 5
Explain the difference between a linear model and a quadratic model.
Line vs. curved; one is symmetrical (quadratic) and one is not (linear)
What is the graphical method for solving systems of equations?
What is the difference between congruent and similar figures?
Congruent figures are exactly the same size and shape.
Similar figures have the same shape but are different sizes.
Describe how to determine the y-intercept from a graph of a linear function.
1. Locate the y-axis: The y-axis is the vertical line on the graph. It represents all points where the x-coordinate is 0.
2. Find the intersection point: Look for the exact point where the graphed line crosses the y-axis.
3. Read the y-coordinate: The y-coordinate of this intersection point is the y-intercept.
What criteria can you use to evaluate the effectiveness of different models in representing data?
A good model is simple and easy to understand.
It should capture the general trend without trying to hit every single point perfectly.
Explain how to determine if a system has one solution, no solution, or infinitely many solutions.
One solution: two lines intersect at exactly one point
No solution: Parallel lines (m is the same) and never intersect
Infinitely many: two equations are the same line and they intersect at every single point along their path.
Describe the process of reflecting a shape over the x-axis.
(x,y)→(x,−y)
Keep the x-coordinate the same.
Change the sign of the y-coordinate.
Given a real-world scenario, how would you set up a linear function to represent that situation? Provide an example.
1. Identify the Rate of Change (the slope, m)
2. Identify the Initial Value (the y-intercept, b)
3. Define Your Variables (x and y)
x is the independent variable, representing the quantity that is changing
y is the dependent variable, representing the total amount or outcome that is being calculated.
When is it appropriate to use an exponential model instead of a linear model?
When there is not a constant growth/decay and/or growth/decay is rapidly or exponentially increasing/decreasing
Solve the system of equations:
2x+3y=6 and x-y=4.
x = 18/5 , y = -2/5
(18/5, -2/5)
How do you determine if two shapes are congruent using transformations?
Congruent means two figures have the exact same size and shape. You can prove that two shapes are congruent if you can map one shape onto the other using a sequence of rigid transformations (all but dilation works).