The Name Game

Learning Focus

Use precise vocabulary to name features of a function: intercepts, increasing/decreasing, rate of change, asymptotes, and domain/range.

Big idea

Key features:

• $y$-intercept: value at $x=0$.
• $x$-intercept (zero): where $f(x)=0$.
• Increasing / decreasing: outputs rise / fall as $x$ increases.
• Rate of change: slope for lines; changing for exponentials.
• Horizontal asymptote: a line the graph approaches but never reaches.
• Domain / range: allowed inputs / resulting outputs.

Linear and exponential functions have different "personalities," and the features tell them apart.

Worked example

Linear $f(x)=2x-6$:

• $y$-intercept: $f(0)=-6$
• $x$-intercept: $2x-6=0\Rightarrow x=3$
• Increasing everywhere, constant rate $2$

Exponential $g(x)=3\cdot2^x$:

• $y$-intercept: $3$
• No $x$-intercept (never $0$)
• Increasing, with horizontal asymptote $y=0$
• Range: $y>0$

Takeaways

• Name the feature, then give its value.
• Linear: constant rate. Exponential: constant ratio and a horizontal asymptote.
• Always connect features back to the context when there is one.