Explore every Algebra 1 function family through interactive investigation. Build deep conceptual understanding โ not just memorisation.
A function assigns to each input exactly one output โ every single time. Think of it as a machine: put something in, get exactly one thing out.
Click any number to feed it into the machine. Watch the output. Could the same input ever give two different outputs from the same function? Why not?
Every input has exactly one arrow leaving it in a function.
If a vertical line crosses the graph more than once, it is not a function.
A function assigns each element of the domain (all valid inputs) exactly one element of the range (all possible outputs). Written f(x), read "f of x".
A graph tells a complete story โ intercepts, increasing/decreasing intervals, maxima and minima. Learn to read all of it without needing a formula.
Hover the graph to read coordinates. Click to pin a point. Can you identify: (a) where f(x) = 0? (b) the highest point? (c) where the function increases vs decreases?
A linear function has a constant rate of change. The slope m tells you how steep the line is. The y-intercept b is where it crosses the y-axis.
โ Drag m from negative to positive โ what changes? โก Set m = 0. What kind of line do you get? โข What does b control? โฃ Find settings that make the line pass through (0, 0).
Slope = Rise รท Run = (yโโyโ)รท(xโโxโ). A linear function in slope-intercept form is f(x) = mx + b. Domain and range are both all real numbers (unless restricted by context).
A quadratic function creates a parabola โ a symmetric U-shape with a turning point called the vertex. The rate of change is not constant.
โ What does a control? Try a = 2 vs a = 0.5 vs a = โ1. โก Where is the vertex? โข What is the axis of symmetry? โฃ When does the parabola open upward vs downward?
Vertex form: f(x) = a(x โ h)ยฒ + k. Vertex at (h, k). If a > 0 range is y โฅ k; if a < 0 range is y โค k. Domain is always all real numbers. Axis of symmetry: x = h.
Exponential functions grow (or decay) at a rate proportional to their current value. The base b controls whether the function grows (b > 1) or decays (0 < b < 1).
โ Set b = 2. By how much does f(x) multiply each time x increases by 1? โก Try b = 0.5. What happens now? โข What is the y-intercept always equal to? โฃ Can an exponential function ever reach zero?
f(x) = a ยท bหฃ where a is the initial value and b is the growth/decay factor. If b > 1: exponential growth. If 0 < b < 1: exponential decay. Domain: all reals. Range: y > 0 (never touches the x-axis โ horizontal asymptote at y = 0).
Absolute value measures distance from zero โ always non-negative. This creates the iconic V-shape. Parameters a, h, and k transform it in predictable ways.
โ Where is the vertex? โก What happens when a is negative? โข What is the range when a < 0 vs a > 0? โฃ What real-world situations create V-shaped graphs?
Vertex form: f(x) = a|x โ h| + k. Vertex at (h, k). If a > 0 range is y โฅ k (opens up). If a < 0 range is y โค k (opens down). Domain is always all real numbers.
A piecewise function uses different formulas for different intervals of the domain. It's like a function that changes its rule depending on where x is.
โ Hover the graph. Which piece applies at the boundary? โก Is the function continuous (no gaps or jumps)? โข What do the open and closed dots tell you? โฃ Can you construct the absolute value function as a piecewise?
โ filled = that point IS included ยท โ open = that point is NOT included. They tell you which piece "owns" the boundary.
The floor function returns the greatest integer less than or equal to x. Written โxโ, it always rounds down toward negative infinity โ never up.
โ Why is โ2.9โ = 2 and not 3? โก Why is โโ0.1โ = โ1 and not 0? โข What is the range of the floor function? โฃ Why is each step closed on the left and open on the right?
A garage charges $3 per hour started. 1.1 hours = 2 hours charged. This uses a ceiling function โ but if you paid only for completed hours, that would be a floor function.
The ceiling function returns the least integer greater than or equal to x. Written โxโ, it always rounds up toward positive infinity.
โ Compare โ2.1โ and โ2.1โ. How do they differ? โก Compare โ3.0โ and โ3.0โ โ are they the same? โข On the graph, which end of each step is closed vs open โ and why is it the opposite of floor?
C(w) = $5 ยท โwโ. A 2.3 lb package is charged as 3 lbs. The company rounds UP (ceiling) to ensure they never undercharge. Each step is open on the left, closed on the right.
Every function family has a distinctive shape, equation form, and domain/range. Learn to recognise and compare all of them.
Straight line. Constant slope. Domain & range: all reals.
Parabola. Vertex at (h,k). Domain: all reals. Range: yโฅk or yโคk.
Curved. Asymptote at y=0. Domain: all reals. Range: y > 0.
V-shape. Vertex at (h,k). Domain: all reals. Range: yโฅk or yโคk.
Different formula per interval. May be continuous or have jumps.
Staircase. Jumps at integers. Domain: all reals. Range: integers โค.
Click a scenario on the left, then click the correct function family on the right.