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Math · 10 · Algebra

Interpretation and Representation

Notes

Key points

Slope = change in y per 1 unit of x, in context units.
y-intercept = value when $x=0$ : starting amount, fixed fee.
x-intercept = when $y=0$ : runs out, breaks even.
In $y=ab^x$ : $a$ = initial value, $b$ = factor per unit of $x$ .
In $3x+5y=60$ : 3 and 5 are per-unit amounts, 60 is the total.
Change of k units in x → change of km in y.
Pick the form that shows the feature: vertex form → max/min, factored → zeros.

Formulas

Slope

m=\dfrac{y_2-y_1}{x_2-x_1}

Units of y per unit of x

Linear model

y=mx+b

m = rate, b = value at x = 0

Change over an interval

\Delta y=m\,\Delta x

Exponential model

y=ab^x

a = value at x = 0

Common traps

Calling a negative slope an increase — “decreases by 4 per year”.
Answering the per-unit change when asked for the change over 5 units.
Swapping x and y: slope is y per x, never x per y.

SAT tip

Plug in x = 0 and x = 1: the first value is the intercept, the difference is the slope.

Test yourself

10 official SAT questions on Interpretation and Representation, easiest first.

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