What this lesson covers
- Translating a function statement into an equation, one slot at a time
- Why stacking the two equations makes b cancel instead of being solved for
- f(0) is always b, and f(1) is always a plus b
- The parametric version, where the unknown moves into the points themselves
- A practice problem, and the three intermediates that get mistaken for the answer
Chapters
- 0:00f(2) = 17 is not a point
- 0:04Two statements, one function
- 0:17The setup: f(x) = ax + b
- 0:37Press each statement into an equation
- 1:22f(0) is b, and f(1) is a plus b
- 1:57Your turn
- 2:37Three ways this goes wrong
- 2:57Two statements, two equations
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:17The setup: f(x) = ax + b
f of two equals seventeen. That is not a point on a graph. That is an equation about a and b. The function is f of x equals a x plus b, and the question hands you two of these statements, then asks for f of zero.
0:37Press each statement into an equation
Translate each statement. f of two equals seventeen means two a plus b equals seventeen. f of five equals thirty two means five a plus b equals thirty two. Two unknowns, two equations. Stack them. Subtract the top from the bottom. b minus b is zero. The b column simply vanishes. Three a equals fifteen, so a equals five. Put a back. Two times five plus b equals seventeen, so b equals seven. And f of zero is a times zero plus b. That is just b. The answer is seven.
1:22f(0) is b, and f(1) is a plus b
Here is the shortcut hiding in the notation. f of zero is always b, the starting value. And f of one is a plus b. Choice C, twelve, is a plus b. It is f of one, a correct answer to a question nobody asked. Sometimes the unknown moves into the points. A line through k comma ten and three k comma twenty two, with slope three. Rise is twelve. Run is two k. Twelve over two k equals three, so k equals two.
1:57Your turn
The function g of x equals c x plus d. g of three is eleven. g of seven is twenty seven. What is g of zero? Translate. Three c plus d equals eleven. Seven c plus d equals twenty seven. Subtract, and d is gone. Four c equals sixteen, c equals four. Then d equals eleven minus twelve. Negative one. g of zero is d. The answer is A.
2:37Three ways this goes wrong
Three ways this goes wrong. Stopping at a equals five, an intermediate is not the target. Reading f of two equals seventeen as slope one, seventeen minus two is not b. And answering a plus b, that is f of one, not f of zero.
2:57Two statements, two equations
Two statements. Two equations. Subtract, and b cancels.
Read the written version: the Linear functions strategy guide, then try 25 hard Linear functions questions with full explanations. Both are free, no account needed.