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∣).
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.
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.
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.
Nice work completing Further Transformations, here's a quick recap of what we covered:
Exercises checked off
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∣).
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.
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.
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.
Nice work completing Further Transformations, here's a quick recap of what we covered:
Exercises checked off