Where this sits. This is one of the five algebra families — a little over half the test, and where most students have the least ground under them. Every family page on this site has the same eight parts, in the same order, so once you know the shape of one you know the shape of all of them. Where a part is empty, it says so.
Start hereThe Coordinate PlaneIf graphs are new, begin with this page: the number line, the two axes, and plotting points, one step at a time, up to the first straight line. Then Lines on the Grid and the quiz, below.Open The Coordinate Plane → DrillsFourteen drills for this family, one skill each, ten problems at a timeGo to the drills ↓A function is a rule that takes one number in and gives one number back, written f(x). The test asks you to work a rule on a number, to find which input gives a stated output, to match a table or a graph to its rule, and to say whether a relationship is a function at all.
Functions are drawn on the coordinate plane, and this family also covers the plane itself: plotting points, the four quadrants, the distance between two points, and the midpoint of a line segment.
What the test asks in this family
- Where a point lives, and the four quadrantsThe Coordinate Plane · Lines on the Grid · Lines and Functions Practice
- The distance between two pointsThe Coordinate Plane · Lines on the Grid · Lines and Functions Practice
- The midpoint of a line segmentThe Coordinate Plane · Lines on the Grid · Lines and Functions Practice
- Function notation, f(x), and what it meansLines on the Grid · Quiz 41 · Lines and Functions Practice
- Working a rule on a number: finding f(5)Lines on the Grid · Lines and Functions Practice
- Finding the input that gives a stated outputLines on the Grid · Lines and Functions Practice
- Matching a table or a graph to its ruleQuiz 41 · Lines and Functions Practice
- Telling whether a relationship is a function at allQuiz 41 · Lines and Functions Practice
All 8 have something on this site.
Six questions across this family. Type an answer and press Check. You get the reasoning for every one, right or wrong.
- f(x) = 3x − 4. What is f(5)? 11. f(5) means put 5 in wherever x appears: 3(5) − 4 = 15 − 4 = 11. The notation looks like multiplication but is not — f(5) is not f times 5. It is a machine with a number going in and a number coming out.
- f(x) = 3x − 4. For which x is f(x) = 20? x = 8. This runs the machine backwards, and it is just an equation to solve: 3x − 4 = 20, so 3x = 24 and x = 8. Questions that give you the output and ask for the input are ordinary equation-solving wearing function notation.
- In which quadrant does the point (−3, 5) lie? The second. The quadrants are numbered counter-clockwise from the top right, so the second is the top left — x negative, y positive, which is exactly this point. Reading the coordinates in the right order is the whole skill: (x, y), across first and then up.
- How far apart are the points (1, 2) and (4, 6)? 5. Go across 3 and up 4, and those two make the legs of a right triangle with the distance as its hypotenuse: 3² + 4² = 25, so the distance is 5. Note that the distance formula is not on the sheet they give you — the line there labelled “distance formula” is d = rt, about rates.
- What is the midpoint of the line segment from (2, 3) to (8, 11)? (5, 7). The midpoint is the average of each coordinate taken separately: x is (2 + 8) ÷ 2 = 5, and y is (3 + 11) ÷ 2 = 7. Halfway across and halfway up. This one is not on the formula sheet either.
- Is this a function: (1, 3), (2, 5), (1, 7)? No. A function gives exactly one output for each input, and here the input 1 is sent to both 3 and 7. One input, two answers — the machine is not reliable, so it is not a function. An output can repeat without a problem; it is a repeated input that breaks it.
Missed one? Quiz 41: One In, One Out in Learn it covers what makes a relationship a function and matching a table or graph to its rule. The Coordinate Plane under Work it by hand teaches the grid, distance and the midpoint from the beginning, Lines on the Grid reviews them with function notation, and Lines and Functions Practice under On paper has the drill. Every part of this family now has a lesson.
Want more than these six questions? Flyover I asks about all fourteen families in one sitting and ends in a grid that says where to start; Flyover II asks them again later, with different questions, so you can see what has moved.
- Quiz 41One In, One OutWhat makes a relationship a function, the vertical line test, and matching a table or a graph to its rule.
One quiz and two rooms, and between them this family is complete. The quiz assumes The Coordinate Plane, in the Start here box at the top of this page, and Lines on the Grid, below. If the grid itself is new, start with the first.
Fourteen short drills, one skill each, in order. Each starts with two worked examples, then gives you ten problems (or five) and the reasoning for every answer. Each one prints, with the answer key on the last page. When you get all ten right, or all five right twice in a row, the drill gets a check mark on the Drills page.
- 14.01Plot a point on the grid
- 14.02Name the coordinates of a point
- 14.03Which quadrant, or which axis?
- 14.04Distance along a grid line
- 14.05Distance between two points (Pythagorean theorem) · calculator allowed
- 14.06The midpoint of a segment
- 14.07Function notation: find f(3)
- 14.08Which input gives this output?
- 14.09Is it a function? Tables, lists of pairs, graphs
- 14.10Match a table to its rule
- 14.11Read a function's graph: values, rising and falling, highest and lowest
- 14.12Linear or not?
- 14.13Compare two functions given in different forms · calculator allowed
- 14.14Domain and range
- The Coordinate Plane · start here if graphs are new: the number line, the two axes, naming and plotting points, the four quadrants, distance and the midpoint, and how points that follow a rule make a straight line. Grids to practice on, and twelve questions at the end.
- Lines on the Grid · a review: where a point lives, the four quadrants, the distance between two points, and the midpoint of a line segment, with worked examples and four sets of questions.
- Lines and Functions Practice · the same sheet as family 13. For this family: the quadrants, distance and midpoint, working a rule on a number, finding the input, telling whether a relationship is a function, and writing the rule from a table. Made to print.
- Khan Academy, math · free videos and practice. Search it for functions, function notation, coordinate plane, distance formula. These are the clips we start from in class.
- Math Antics · short, plain videos on the basics.
- Light & Salt Learning · free GED classes online, with playlists by topic.
- Get Sum Math · GED math on video, one topic at a time.
- Kaplan GED Test Prep and the McGraw-Hill Mathematical Reasoning Workbook · look up functions and the coordinate plane in the contents of either.
That is all fourteen families. Two things are worth doing now. Take Flyover II — the same fourteen families, fifty-five new questions — and see what has moved since the first time. And, if your test date is close, sit the free full-length practice test at GED.com under the clock, with the calculator and the formula sheet, as on the day.