Algebra functions are taught beginning in Algebra I courses and continued in Algebra II courses. They are formulas that express a math rule or relationship between two variables, typically x and y, in which x is considered an independent variable and f(x) or y is the dependent variable. The equation for the relationship represents a line on a coordinate plane.
Your opponent wants to know if you understand there can be only one value for y for any given value of x. |
This means that if the line curves and one part of the line is above another part, it is not a function. However, it can bend and be beside itself and still be a function. Think of a parabola. If it is open to the left or right, it can’t be a function. If it opens up or down, it can be a function.
Your opponent wants to make sure you understand the terms domain and range. |
The domain of the function or formula is the set of all possible values for which the function is defined. That means when you plug x into the function, the solution has to be a real number. (There are no imaginary numbers on the SAT or ACT.) The range is the set of values you get when you apply the function or rule.
Your opponent is going to use some standard terminology related to functions. |
ƒ(x) means “function of x.” It is really a fancy substitution for y in a formula. For example:
The relationship, ƒ(x) = x^{2} + 1, means about the same thing as y = x^{2} + 1. In the first case, it is a range of values (domain) that are possible for x, usually, -∞ ≤ x ≤ ∞. In the second case, you usually see just one value for x.
When you see the function notation make sure you don’t confuse it with multiplication. ƒ(x) does not mean “f times x.”
“Evaluating a function” just means you need to solve the function (equation) for a particular value of the value that is inside the parentheses. So evaluate the function ƒ(3) = x^{2} + 1 means to solve the equation by substituting 3 for x; the solution is 10.
A really good website that will help you review functions and their definitions is:
http://tutorial.math.lamar.edu/classes/alg/functiondefn.aspx
Graphs of a Function
Your opponent want to see if you understand functions are graphs and how they may be moved around on the coordinate grid. |
“The effect of a translation on a graph” refers to how the graph will be moved or shifted on the coordinate grid by a simple change in the function. On the SAT and ACT, a graph can be moved left, right, up, or down from its original position. It’s not so difficult if you follow a couple of procedures.
If the change is in the right side of the equation, then the graph moves or slides up with positive values and down with negative ones. For example, in these two equations:
Equation #1: ƒ(x) = x^{2}
Equation #2: ƒ(x) = x^{2} – 1
the only difference is the –1. Think about what the second equation does to f(x) or the value of y for a given value of x. It makes it smaller by subtracting 1. That is going to shift the values of the function down 1 place on the coordinate grid. The opposite translation would occur if it were positive.
The other possible case is when the change is inside the parentheses of the function. Consider the following two equations.
Equation #3: ƒ(x) = x^{2}
Equation #4: ƒ(x+1) = x^{2}
By placing the change inside the function notation, the graph is shifted left or right. It is moved to the left for positive values and the right for negative values. In the case of Equation #4, the graph is the same as it is in Equation #3 except it is moved one position to the left.
Take a look at http://library.thinkquest.org/20991/alg2/quad2.html#transform for more information. You only need to review the “translations” part of the page. The other sections aren’t on the SAT or ACT!
Intercepts for the x and y-axes
Your opponent expects you to understand how to determine where a line intercepts the x or y axis. |
You could see a graph that goes up and down like a roller coaster or a V (or M) or an upside down V (or W). In these cases, you would have more than one place on the graph where y would have the same value. However, notice the value of x would be different. If you see a question such as “For how many values of x does f(x)=2?”, just take a look at the graph and see how many times the line goes through y=2.
http://www.wingo.com/tides/index.html
In this example, you see the height of the tide in Santa Cruz, California at different times of the day. Please note this is not a mathematical function like you will see on the SAT and ACT (since tides are controlled by the position of the sun and moon) but it shows a line that intercepts the value of y=2 at four places: 3 and 11 on the date 05-21 and 4 and 12 on the following day. This example should help you understand that on SAT and ACT questions, you will sometimes see questions that look like complicated functions but you simply should remember this graph of the tides. It represents how a math function operates—there is only one value of y (tidal depth) for any value of x (time of day). You can see in this example that there are four values of x for one value of y.
If you get an x– or y-intercept question, simply set the value to 0 and solve the equation or read the graph. Use what you know (Pillar III); the y-intercept occurs when x is zero and vice-versa. If the question is about the x-intercept for an equation, just set y or f(x) to 0 and solve. For example:
ƒ(x) = x^{2} + x – 12
y = x^{2} + x – 12
0 = x^{2} + x – 12
0 = (x + 4) (x – 3)
x = -4, x = 3
This means that when y or f(x)=0, then x is both 3 and –4. Those are the x-intercepts.
Quadratic Equations:
Quadratic equations (ƒ(x) = ax^{2} +b x -c) are always in the shape of a parabola. Here’s some things to remember about the equation:
- If the value of a is positive the parabola opens up
- If the value of a is negative the parabola opens down
- The greater the value of a, the greater the slope and the narrower the parabola
- The maximum/minimum is the high/low point of the parabola (vertex)
- The x-coordinate of the vertex equals -b/2a
- The y-coordinate of the vertex equals (b2-4ac)/4a
- ƒ(x) = a(x-h)^{2} +k is referred to as the vertex form where h and k are the x- and y-coordinates respectively.
Reflections of a Function
You could see a question asking what the equation of a reflection of an original equation would be. For example: Line p is the graph of the linear equation y = x + 2. Line r is the reflection of line p in the x-axis. What is the equation of line r?
This sounds much more complicated than it is. Line r is above the x-axis and its reflection or mirror image, line r, is below the x-axis. In fact, it is exactly as far below the axis as p is above. That means all we need to do is change the signs of x and 2. The answer is y = -x – 2 . If you see a question like this on the SAT or ACT, just sketch a graph of it on a grid. Then draw its mirror image. You will quickly see the relationship.
Here’s the way you probably saw this issue described in your algebra class:
If y = f(x), then
y = f(−x) is its reflection about the y-axis,
y = −f(x) is its reflection about the x-axis.
There’s more about this issue later when we discuss the slope-intercept equation.
When a function is middle school math
Your opponent will see if you panic if she makes an easy calculation look like a function. |
Sometimes the test writer makes a question look a lot harder than it is. Remember Pillar VI: Don’t Be Intimidated. You may see a question that is explained as a function and then values are given for the variables in the equation. You just need to substitute the values and turn the math crank. Don’t be daunted (frightened) by the fact it looks tough. Just do it; it’s as simple as putting on your Nike sneakers! For example:
The daily cost c, in dollars, of driving the Wizard’s motor home m miles is represented by the function c (m) = .75m + 16n where n is the number of days he has the fixed daily cost of expenses such as payments and insurance. It took him two days to drive his motor home to the Wizard Conference. He calculated his cost to be $782. How many miles did he drive to the conference?
Answer #1
Slope-Intercept Equation
Your opponent wants to know if you understand and can apply the slope-intercept equation. |
You can tell the slope and y-intercept of the line of an equation when the equation takes the following form:
y = mx + b
The slope of the line is indicated by m. Be careful of whether it is positive or negative.
The line crosses the y-axis (the y-intercept) based on the value of b. Again make sure you pay attention to whether it is positive or negative.
For example, the equation y = .5x + 3 indicates the slope for the line is ½ (it rises 1 while it runs 2) and it intercepts the y-axis at +3. That is where x=0.
Use the slope-intercept equation to better understand reflections of a line in a coordinate grid. If you reflect in the x-axis, the slope is the negative value and the intercept is also going to be the negative of the original y-intercept. If a line is reflected in the y-axis, the slope again becomes the negative value, however, the y-intercept remains the same.
A final important fact to remember is that when two straight lines on the coordinate grid are at a right angle (perpendicular) to one another their slopes are the negative reciprocals of one another. For example, if one of the lines has a slope of 3 the slope of the other is –1/3.
Sample Questions
5. The graph of which of the following equations is perpendicular to the graph of 2y = 2x + 1?
(A) y + x = 7
(B) 2y – 2x = 1
(C) 2y = x + 1
(D) y = x + .5
(E) 2y = 2x + 2
6. What are the (x,y) coordinates of the unique point on the graph of 2x + 3y = -9 when y = -5?
(A) -5, 3
(B) -5, 0
(C) 2, -5
(D) 3, -5
(E) 3, 5
15. Which of the following functions most closely represents the graph above?
(A) f(x) = x^{2} +3
(B) f(x) = (x^{2} – 3) -1
(C) f(x+3) = x^{2} – 1
(D) f(x) = x^{2}-1
(E) f(x) = (x + 3)^{2} + 1