Functions

In this lesson you will: Name the family — linear, exponential, or quadratic — from a table, a graph, or a sentence; Read key features from whichever representation you are handed, and interpret them in context; Treat a system, an intersection, and the equation f(x)=g(x) as three names for one question.

Name the family first, then everything else follows

Almost every question in this category starts with one decision, and students who make it quickly finish the item. Given equal steps in the input, look at what stays constant in the output.

Constant differences mean linear. Constant second differences mean quadratic. Constant ratios mean exponential. A sequence is the same idea with the inputs restricted to whole numbers.

Key features, and which form shows them

A quadratic has the same graph whichever way you write it, but each form hands you a different feature without work. Factored form shows the zeros. Vertex form shows the maximum or minimum. Standard form shows the y-intercept.

So the real question on a F.IF.8 item is not \"can you rewrite this\" but \"which rewrite answers what was asked.\" Read the question, then choose the form.

For an exponential, the y-intercept is the coefficient, the base is the per-period factor, and the horizontal asymptote is the line the graph flattens toward.

Systems: one question wearing three costumes

\"Solve the system,\" \"where do the graphs meet,\" and \"solve f(x)=g(x)\" are the same task. The answer is the input where two outputs agree, and the point on the graph where two curves cross.

Elimination is legal for a reason worth being able to state: replacing one equation with the sum of it and a multiple of the other produces a system with exactly the same solutions, because every pair that satisfies both originals satisfies the combination, and the step can be undone.

For inequalities, the boundary line is dashed when the sign is strict and solid when it is not, and the shading is the half-plane containing every point that makes the statement true. Test one point that is not on the line.

Common mistake — Reading a growth factor as a growth rate

Given f(t)=400(1.06)^t, a student says the quantity grows by 106% a year. Given g(t)=400(0.94)^t, the same student says it falls by 94%.

Both readings take the base straight out of the expression and attach a percent sign to it. That is exactly what the expression looks like it is saying.

But the base is the factor the quantity is multiplied by, not the amount it changes. 1.06 keeps all of it and adds 6%. 0.94 keeps 94% of it, which is a 6% loss.

How to catch it: Test with a round number. Start at 100. One year later 1.06 gives 106 and 0.94 gives 94 — changes of 6, not 106 or 94. If the base is above 1, subtract 1 for the rate; if it is below 1, subtract it from 1.