SAT Linear Functions: The Complete Guide

For a table, divide changes in output by changes in input. A table with inputs 1, 4, and 9 and outputs 8, 20, and 40 has rates $12/3=4$ and $20/5=4$. The output differences differ because the input intervals differ. Equal slopes support the linear model; looking only at the differences would wrongly reject it.

Once you know $m$, substitute any known pair into $y=mx+b$ to find $b$. Alternatively, use point-slope form $y-y_1=m(x-x_1)$. With slope four through $(1,8)$, this gives $y-8=4(x-1)$, which expands to $y=4x+4$. Both forms define the same outputs.

Standard form $Ax+By=C$ can also describe a linear function when $B\ne0$: solving for $y$ gives $y=-\frac{A}{B}x+\frac{C}{B}$. A vertical line, where $B=0$ and $A\ne0$, does not define $y$ as a function of $x$ because its one input corresponds to many outputs. A horizontal line does define a function: its slope is zero.

In $f(2x-1)$, replace every occurrence of the original input with the whole expression $2x-1$. If $f(u)=5u+6$, then $f(2x-1)=5(2x-1)+6=10x+1$. Using a temporary letter such as $u$ can make the input substitution clearer.

An output transformation $g(x)=f(x)+k$ adds $k$ to every height and leaves slope unchanged. An input shift $g(x)=f(x-h)$ moves the graph right by $h$. For a line, this becomes $mx+(b-mh)$, so the new intercept changes by $-mh$, not automatically by $-h$.

A horizontal function deserves the same interpretation routine as a sloped one. If $f(x)=23$, every permitted input has output 23, the slope is zero, and the vertical intercept is 23. It is linear in the constant-rate sense used here. It is not proportional unless that constant is zero. You cannot reverse this function into a unique input rule: many inputs give the same output.

An input equation can therefore have special outcomes. For the constant function above, $f(x)=23$ holds for every allowed input, while $f(x)=24$ holds for none. Before dividing by a slope to find an input, confirm that the slope is nonzero.

Desmos can check a model by graphing its equation and entering the given points. Every point must lie on the proposed line. If you enter a small data table, plotting the points can expose a sign error or a missed intercept quickly. Exact two-point slope arithmetic is usually enough to determine the line without a fitting procedure.

To find when a model reaches a target $T$, graph $y=f(x)$ and $y=T$. Read the intersection's horizontal coordinate. To compare two models, graph both and inspect where they intersect. The horizontal coordinate gives the shared input; the vertical coordinate gives the common output. A question asking for the shared cost needs the second coordinate.

Use numerical scales carefully. A steep-looking line is not necessarily changing faster than a line in another graph if axis units differ. Interpret slopes using coordinate differences and units, not the angle a line appears to make on your screen.

Do not confuse an intercept with an intercept's coordinate. A $y$-intercept of 30 corresponds to the point $(0,30)$. The $x$-intercept is found by setting the output to zero. A question asking when a balance reaches zero needs that input, not the initial balance.

The phrase increases by five each hour describes an additive rate. Increases by five percent each hour describes a multiplicative rule and generally leads to an exponential model. A line can match two points from either situation, so use the described mechanism as well as the numbers.

When tables use uneven input intervals, calculate rates rather than compare raw differences. When a question changes units from minutes to hours, convert the rate too. A rate of three units per minute equals 180 units per hour; changing only the variable label makes a different function.

Keep the domain in the answer. If solving a continuous model gives 7.4 buses for a transportation count, that is not automatically the final count. Determine whether the setting requires rounding up to meet a capacity requirement or whether the model was intended for a continuous quantity.

A difference of outputs often reveals slope without revealing the intercept. If $f(9)-f(4)=30$, then $5m=30$, so $m=6$. There is still not enough information to find $b$. You can determine every output change from this information, but you cannot determine an absolute output such as $f(0)$ without another condition.

A change in input can alter the rate with respect to a new variable. If $C(h)=14h+32$ and $h=3d$, then $C(3d)=42d+32$. The starting charge stays 32, while the slope is now 42 dollars per unit of $d$. Read slope relative to the variable actually used on the horizontal axis.

To compare $f(x)=mx+b$ and $g(x)=nx+c$, subtract them: $f(x)-g(x)=(m-n)x+(b-c)$. Equal slopes make the difference constant. Different slopes make the gap change linearly. Setting the difference to zero finds a crossing; evaluating its sign tells which model is greater at a chosen input.

A reciprocal rate can appear when the input and output trade roles. If distance is $d=60t$, then time is $t=d/60$. The original slope is 60 miles per hour, and the inverse relationship has rate one sixtieth of an hour per mile. The units explain why the coefficient must change.

Plan a 22-minute session. Spend five minutes interpreting slopes and intercepts without calculating any missing values. For each model, say the input unit, output unit, per-unit rate, and meaning of input zero. If the model has a restricted domain, identify it before evaluating.

Use the next eight minutes in Solar on linear-function questions presented as equations, tables, and word problems. Write the representation you were given and the representation you built. Spend four minutes on composed inputs such as $f(2x+3)$ and output differences where the intercept cancels.

Reserve five minutes for correction. Rebuild each missed model from its units, then check two points. If you used Desmos, explain which coordinate answered the question and why. If the graph only confirmed an answer you already understood, note whether it saved time or merely added typing.

On a later mixed set, compare two methods on one question: a full equation and a direct change calculation. Choose the method you can explain and verify fastest. Familiarity should shorten sound reasoning, not replace it.