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Unit 1
30–40% of examThis unit is about how two quantities change together, and about two big function families: polynomials and rational functions. You'll read rates of change from graphs and tables, find zeros, asymptotes and holes, and choose and build functions to model real data. These skills come back in every later unit and in calculus.
Longer videos that cover the whole unit. Good for a first pass or a final review.
A function pairs each input with exactly one output, so you can track how the output changes as the input changes. You'll describe where a graph is increasing or decreasing, where it's concave up or down, and where it crosses the x-axis.
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The average rate of change over an interval is the change in output divided by the change in input. Using tiny intervals around a point, you can estimate how fast the function is changing right at that point.
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A linear function has the same average rate of change on every interval, so its graph is a straight line. For a quadratic, the average rates over equal-length intervals change by the same amount each time, which is why its graph curves.
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A polynomial is a sum of terms like a·xⁿ with whole-number powers, and its degree is the highest power. You'll find where a polynomial turns from rising to falling, giving local (relative) and global (absolute) maximums and minimums, and spot points of inflection where concavity changes.
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Counting repeats, a degree-n polynomial has n zeros in total, and some may be non-real complex numbers like 2 + 3i. When the coefficients are real, non-real zeros come in conjugate pairs, so 2 − 3i is a zero too. A real zero's multiplicity tells you whether the graph crosses the x-axis there (odd) or just touches it (even). You'll also test whether a function is even (mirror image across the y-axis) or odd (symmetric about the origin).
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End behavior is what the graph does as x → ∞ and x → −∞. For a polynomial it depends only on the degree (even or odd) and the sign of the leading coefficient.
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A rational function is one polynomial divided by another. Far from the origin, the ratio of the two leading terms decides whether the graph levels off at a horizontal asymptote, heads toward a slant asymptote, or grows without bound.
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A rational function equals zero where its numerator is zero and its denominator is not. You'll find zeros from factored form and use sign charts to see where the function is positive or negative.
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A vertical asymptote appears at an x-value that makes the denominator zero when that factor doesn't fully cancel with the numerator. Near it, the outputs grow without bound, toward ∞ or −∞.
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If a factor in the denominator cancels completely with the same factor in the numerator, the graph has a hole there instead of an asymptote. You find the hole's height by plugging the x-value into the simplified function.
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The same function can be written in standard form, factored form, or as a quotient plus remainder, and each form makes different features easy to see. You'll use polynomial long division to find slant asymptotes and the binomial theorem (Pascal's triangle) to expand powers like (a + b)ⁿ.
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Adding or multiplying inside or outside a function shifts, stretches, shrinks or reflects its graph. For example, g(x) = a·f(b(x − h)) + k shifts f right by h and up by k, stretches or shrinks it vertically by a factor of |a| and horizontally by a factor of 1/|b|, and a negative a or b also reflects it.
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To model data or a situation, you choose a function type that matches its pattern of change, such as linear for a constant rate or quadratic for a rate that changes steadily. You also state the assumptions and limits of your model, like a sensible domain.
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Once you've chosen a model type, you build it from data or given information, often with a calculator regression, then use it to predict values and answer questions in context. You also judge when the model's answers make sense.
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