Mistake Master
Reading lines and functions
Linear functions are the SAT Algebra domain at its most graphical. The math is quick once you read the right thing: the slope, not the intercept; a perpendicular slope with both moves done; the output of a function at an input, not a product. Read the feature the question names, then apply the rule exactly.
§1
What this topic is about
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A linear function is a straight line with a rule behind it: $y = mx + b$ turns an input into an output at a constant rate. The work splits in two: read the right feature off the equation or the graph, then apply the rule the question names, whether that is a perpendicular slope or a function value. The arithmetic is short. The points leak in reading the wrong feature, and in applying a rule only halfway.
§2
Slope and intercept live in different places
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In slope-intercept form $y = mx + b$, the number multiplying $x$ is the slope $m$, and the lone constant is the $y$-intercept $b$. The slope is how steeply the line climbs or falls; the intercept is where it crosses the $y$-axis, the value of $y$ when $x = 0$. Grabbing one when the question wants the other is the most common slip in this topic.
- The slope is the coefficient of $x$, sign included. In $y = -4x + 3$, the slope is $-4$, not $4$.
- The $y$-intercept is the constant term, the height at $x = 0$. In $y = -4x + 3$, it is $3$.
- On a graph, the slope is rise over run; the $y$-intercept is the point where the line meets the $y$-axis.
§3
Slope between two points, and from a graph
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Between two points, the slope is the change in $y$ over the change in $x$. Subtract in the same order on top and bottom. On a graph, count the rise over the run and keep the sign of the rise: a line that falls to the right has a negative slope.
- The formula is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Pick a start point and stay consistent.
- Rise over run, not run over rise. Flipping them gives the reciprocal of the slope.
- A horizontal line has slope $0$; the rise is zero no matter the run.
Worked example. A line passes through $(1, 2)$ and $(4, 11)$. Find its slope.
From $(1, 2)$ to $(4, 11)$, $y$ changes by $11 - 2 = 9$ and $x$ changes by $4 - 1 = 3$.
The slope is $$m = \dfrac{9}{3} = 3.$$ Subtracting in opposite orders, $\dfrac{2 - 11}{4 - 1}$, flips the sign and gives $-3$.
§4
Parallel and perpendicular slopes
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Two lines are parallel when they share a slope. They are perpendicular when their slopes are negative reciprocals: flip the fraction and change the sign. Both moves have to happen. Doing only one is the slip this section is about.
- Parallel lines have equal slopes. A line parallel to one with slope $\dfrac{3}{4}$ also has slope $\dfrac{3}{4}$.
- Perpendicular slopes are negative reciprocals: flip $\dfrac{3}{4}$ to $\dfrac{4}{3}$, then negate to $-\dfrac{4}{3}$.
- Treat a whole number as a fraction over $1$: the perpendicular of $4 = \dfrac{4}{1}$ is $-\dfrac{1}{4}$.
Worked example. Line $\ell$ has slope $-\dfrac{2}{5}$. Find the slope of a line perpendicular to it.
$-\dfrac{2}{5}$ becomes $-\dfrac{5}{2}$.
$-\dfrac{5}{2}$ becomes $$\dfrac{5}{2}.$$ Flipping without negating leaves $-\dfrac{5}{2}$; negating without flipping leaves $\dfrac{2}{5}$. Each is half the rule.
§5
Function notation is a lookup, not a product
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The notation $f(3)$ means the output of $f$ when the input is $3$. It is not $f$ times $3$. To evaluate, put the input in for $x$ everywhere it appears, then simplify. A constant function like $f(x) = 8$ returns the same output for every input.
Worked example. The function $f$ is defined by $f(x) = 10 - 2x$. Find $f(3)$.
$f(3) = 10 - 2(3)$.
$$10 - 6 = 4.$$ Reading $f(3)$ as $3$ times the output gives $12$; treating $-2x$ as $+2x$ gives $10 + 6 = 16$.
Read the slope and intercept in their own places, do both moves the perpendicular rule asks for, and treat $f(a)$ as the output at an input, not a product.
§6
Three patterns that cost real points
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Three patterns recur on linear functions. They are the same ones the diagnostic routes on.
Slope and intercept get mixed up.
The number on $x$ is the slope; the lone constant is the $y$-intercept. Reading $y = -4x + 3$ as slope $3$, or the intercept as $-4$, answers about the wrong feature. On a graph, the steepness and the crossing point get swapped.
Fix. Name which one the question wants, slope or intercept, then read the matching place: the coefficient of $x$ for slope, the constant for intercept.
A perpendicular slope is only half-changed.
The fraction gets flipped but not negated, or negated but not flipped, or the original slope gets reused as if the lines were parallel. Each leaves the rule half-applied.
Fix. Do both moves, flip and negate, and treat a whole number as a fraction over $1$ before flipping.
A function value is read as a multiplication.
$f(3)$ becomes $f$ times $3$, or the input gets added to the output, or the input is echoed back as the answer. The parentheses hold an input, not a factor.
Fix. Substitute the input for $x$ and simplify. $f(3)$ is the output at $3$, a lookup, not a product.
Ten quick checks across the three patterns: reading slope and intercept, the perpendicular rule, and function notation as a lookup. Pick or type your answer, then check. Progress is saved.