Precalculus: families of functions · Year 11
In Precalculus: families of functions (Year 11) you practise 2 skills: Translating a graph · Reflecting a graph. The domain of a function is every value you are allowed to put in. Under a square root the expression is not negative, and in a fraction the denominator is not zero. An even function stays the same when x is replaced by its opposite, and its graph is symmetric in the y axis. An odd function changes sign, and its graph is symmetric about the origin.
Adding a number to a function moves its graph up or down, and adding a number to x inside the function moves it left or right, in the direction opposite to the sign. Multiplying the function by minus one reflects the graph in the x axis, and replacing x by its opposite reflects it in the y axis. Multiplying the function by a number greater than one stretches the graph away from the x axis. The absolute value of a function keeps the parts above the x axis and reflects upwards the parts below it. Where a question has a picture, it shows both graphs: the original dashed and the new one solid.
Tip: Check one point: take a point on the original graph and see where it goes. That is enough to know which way the graph moved.
What you learn
Translating a graph
For example: The point (5, −4) is on the graph of y = f(x) . Which point matches it on the graph of y = f(x) + 3 ?
Answer and solution
The answer: (5, −1)
The y coordinate goes up by 3, and the x coordinate stays the same: (5, −1)
Reflecting a graph
For example: The point (5, −4) is on the graph of y = f(x) . Which point matches it on the graph of y = −f(x) ?
Answer and solution
The answer: (5, 4)
The y coordinate changes sign, and the x coordinate stays the same: (5, 4)
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).