Basics of Functions
A relation is a function if each input (x) maps to exactly one
output (y)
In a distance-versus-time graph, what does a horizontal segment mean?
A: The object is not moving; distance is constant.
Define “increasing” and “decreasing” on a coordinate plan in terms of x and y.
Increasing: y rises as x increases; Decreasing: y falls as x increases.
Match family → shape: Linear, Quadratic, Exponential, Absolute Value (choices: line, parabola, growth/decay curve, V-shape).
Linear → line; Quadratic → parabola; Exponential → growth/decay curve; Absolute Value → V-shape.
Most non-vertical lines have domain = __________ and range = __________.
Domain: all real numbers; Range: all real numbers
Name the quick graph test that determines if a relation is a function.
A: Vertical Line Test
Describe what a line with positive slope communicates about how the quantity changes over time.
The quantity increases at a constant rate as time increases.
A parabola has vertex (−1,4) and opens downward. Is the vertex a maximum or minimum? Explain.
Maximum; opening down makes the vertex the highest point
Write a linear function in slope–intercept form and identify slope and y-intercept.
A: y=mx+b; slope m, y-intercept b.
For y=(x-3)^2+2 give domain and range in words.
A: Domain: all real x-values; Range: y-values at or above 2.
If f(x) = 2x – 5, evaluate f(4) and f(0).
f(4) = 3; f(0) = –5
Water is drained at a constant rate from a full tank.” Describe the graph’s general shape and appropriate axes.
A straight decreasing line; x-axis = time, y-axis = volume/height of water.
State the rule for the Vertical Line Test and what failing it implies.
A: If any vertical line hits a graph more than once, it fails the test and is not a function.
An absolute value function has vertex (1,−3). State a possible equation and name the family.
y=a∣x−1∣−3 (e.g., y=∣x−1∣−3); family: Absolute Value.
An absolute value graph has vertex (1,−2) and opens upward. State its domain and range in words.
A: Domain: all real x-values; Range: y-values at or above −2
A table lists x=3x = 3x=3 twice with two different y-values. Is it a function? Why/why not?
Not a function; one input maps to two outputs.
When x increases by 1, y increases by 4 every time. Linear or nonlinear? Justify.
Linear; constant rate of change (slope = 4).
Name and describe two types of symmetry a graph might have and how to recognize them.
Vertical (mirror across a vertical line x=h); point/rotational (symmetric about a point, e.g., origin for odd functions).
For y=a⋅b^x, when is it increasing and when is it decreasing? (what value changes)
Growth if b>1; decay if 0<b<1.
A function models time since start (minutes) and height of water (inches). Give realistic domain and range restrictions
Domain x≥0 (time nonnegative); Range y≥0, capped
Explain the difference between discrete and continuous functions and give a real-life example of each.
Discrete: separate, countable points (e.g., number of people). Continuous: unbroken values over an interval (e.g., height over time).
From a verbal description alone, name two details you need to graph accurately and why.
Examples: initial value (starting point) and units/scale (to interpret/change correctly). Others: time window (domain), constraints.
Challenge: Both teams can get points for this question.
Teams have 30 seconds to write as many characteristics as possible-- only one person can write on the board from each team but other team members can give answers.
A parabola opens down with vertex (2,5). Give a possible quadratic in vertex form and explain what a controls.
y=a(x−2)^2+5 with a<0 (e.g., y = -(x - 2)^2 + 5); a controls direction (sign) and width/steepness (magnitude).
A piecewise function is defined only on 0≤x≤60 and outputs stay between −1 and 4 State domain and range; name one interval where it could be increasing.
Domain: 0≤x≤60 ; Range: −1≤y≤4-1. Increasing on any sub-interval where y rises as x increases (e.g., 2<x<42, if defined that way).