Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
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Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Transformations & asymptotes
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
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Working on it
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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The graph of y=f(x)+b can be obtained from the graph of y=f(x) by a vertical translation b units upwards (if b<0, the transformation may also be called a translation ∣b∣ units down).
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The graph of y=f(x−a) can be obtained from the graph of y=f(x) by a horizontal translation a units to the right (if a<0, the transormation may also be called a translation ∣a∣ units to the left).
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If a point P is translated by a vector (ab), apply a translation a units to the right and b units up:
The graph of y=pf(x) can be obtained from the graph of y=f(x) by a vertical stretch with scale factor p.
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The graph of y=f(qx) can be obtained from the graph of y=f(x) by a horizontal stretch with scale factor q1.
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The graph of y=−f(x) can be obtained from the graph of y=f(x) by a reflection in the x-axis.
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The graph of y=f(−x) can be obtained from the graph of y=f(x) by a reflection in the y-axis.
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The graph of y=f(ax+b)=f[a(x+ab)] can be obtained from the graph of y=f(x) by a horizontal translation b units to the left, followed by a horizontal stretch with scale factor a1.
OR a horizontal stretch with scale factor a1 followed by a horizontal translation ab units to the left.
The reciprocal function is defined by f(x)=x1.
Notice that f(x) is not defined for x=0. In fact, since x1 gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.
And since for very large x, x1 approaches zero, there is also a horizontal asymptote y=0.
Notice also that x11=x, so f(x)=x1 is self-inverse.
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A linear rational function has the form
When the denominator is zero the graph will have a vertical asymptote:
And as x gets very large, the +b and +d can be ignored:
So there is a horizontal asymptote at y=ca.
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When a rational function is of the form
there is a vertical asymptote at
By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation
for some constant c determined during the polynomial division.
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When a rational function is of the form
There will be vertical asymptotes when the quadratic cx2+dx+e=0.
The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.
Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.
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Note that if the numerator and denominator share a root (ie x=−ab is a root of the denominator), then there will only be one asymptote and a "hole" on the x-axis. This has never shown up on exams.
To obtain the graph of f(∣x∣), you can think of it as:
For x≥0: ∣x∣=x, so f(∣x∣)=f(x).
For x<0: ∣x∣=−x, so f(∣x∣)=f(−x).
Practically, you take the portion of y=f(x) for x≥0 and reflect it across the y-axis to fill in the x<0 side. The right side of the original graph becomes the entire graph of f(∣x∣).
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To obtain the graph of ∣f(x)∣, take the graph of f(x) and:
Leave all points where f(x)≥0 as they are, since ∣f(x)∣=f(x) in that region.
Reflect any parts of the graph where f(x)<0 above the x-axis, because ∣f(x)∣=−f(x) whenever f(x) is negative.
Effectively, every negative y-value becomes positive, mirroring the portion of the curve below the x-axis to above it.
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Notice the "sharp corners" where the function touches the x-axis.
To sketch [f(x)]2 from f(x):
Square all y-values: for each x, the new y-value is (f(x))2.
Because y2≥0, the entire graph of [f(x)]2 is on or above the x-axis.
Points where f(x)=0 remain on the x-axis.
Any minima below the x-axis become maxima above the x-axis.
Any maxima below the x-axis become minima above the x-axis.
If ∣f(x)∣<1, squaring makes it closer to 0; if ∣f(x)∣>1, squaring makes it larger.
Negative parts of f(x) become positive when squared, so everything below the x-axis is reflected above it, with the magnitude of y-values adjusted according to the square.
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Notice the "rounded corners" where the function touches the x-axis, as opposed to the sharp corners for ∣f(x)∣.
To sketch f(x)1 from f(x):
At every x-value, take the reciprocal of the original y-value: y→y1.
Points where f(x)=0 become vertical asymptotes, since 01 is undefined.
If ∣f(x)∣ is large, then f(x)1 is small (close to the x-axis). Thus, vertical asymptotes of f(x) become horizontal asymptotes at y=0. Conversely, if ∣f(x)∣ is small, then f(x)1 is large.
Local maxima and minima "flip":
a maximum at (a,b) on f(x) becomes a minimum at (a,b1) on f(x)1;
a minimum at (c,d) becomes a maximum at (c,d1). This happens because reciprocal values invert magnitudes.
The absolute value of x is defined as
Inequalities of the form
can be solved either algebraically or with technology.
It is crucial to remember that when multiplying both sides of an inequality by a negative number, the inequality changes direction: