Sketching and Comparing Functions
Some graph questions have no numbers at all: sketch a graph of someone's walk to school, or of water draining from a tub. These qualitative graphs only need the shape to be right. Others hand you two functions in different costumes — one as a table, one as an equation — and ask which grows faster. Both skills come down to translating between a description and a picture.
Three shape rules cover the sketching side: steady change draws as a straight slanted segment, no change draws as a flat segment, and faster change draws steeper. For the comparing side, you compute the same two numbers from each function — the rate of change and the -intercept — no matter what form each function arrives in.
Sketching from a story
Walk through the story piece by piece and give each piece its own segment. Someone walking at a steady pace produces a straight, slanted segment. Someone standing still produces a flat segment — time keeps passing, but the distance stays put. Someone jogging produces a steeper segment than someone walking, because the distance changes faster.
Direction matters too. If the vertical axis is distance from home, walking away slants the graph upward and walking back home slants it downward, returning to zero.
The graph below sketches a trip like this: a steady walk (a slanted segment), then a pause (a flat segment while time keeps passing), then a faster stretch (a steeper segment).
Comparing rate of change
The rate of change measures how much the output changes for each -unit change in the input. From an equation in the form , the rate is the coefficient — for , the rate is . From a table, pick two rows and divide: rate .
Watch the input column in tables. If the -values step by , an output jump of per row is a rate of , not — divide the output change by the input change every time.
Comparing -intercepts
The -intercept is each function's output at . From , it is the constant, . From a table, find the row where and read the output. From a graph, read the height where the curve crosses the y-axis. Once both functions give you a number, the comparison is just which number is bigger — a greater -intercept means that function starts higher.
Worked examples
Example 1: sketch a trip to the library
Maya walks to the library at a steady pace, stops at a crosswalk, then walks faster the rest of the way. Sketch her distance from home over time.
Answer: Three segments: rising, flat, then rising more steeply
Example 2: compare rates from a table and an equation
The table for gives , , . The equation is . Which function has the greater rate of change?
Answer: Function — its rate of change is , compared to for
Example 3: a table that steps by 2
The table for gives , , . Is the rate of change of greater than the rate of ?
Answer: No — changes by per unit, while changes by
Try one yourself
Common questions
What does a flat section of a graph mean?
The output is not changing while the input keeps going. On a distance-time graph, a flat section means the person is not moving — time passes, but the distance from home stays the same.
How do I compare functions written in different forms?
Convert both to the same two numbers: the rate of change and the -intercept. An equation hands them to you as and ; a table makes you compute the rate from two rows and read the row; a graph makes you read them off the picture. Then compare number to number.
Why can't I just compare how much the outputs jump in a table?
Because the inputs might not step by . A table stepping by with outputs jumping by has a rate of . Always divide the change in output by the change in input: .
Does a greater -intercept mean the function grows faster?
No. The -intercept is where the function starts; the rate of change is how fast it grows. A function can start higher and still be overtaken — that is exactly what happens when its rate is smaller.
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