Sums and differences of graphs

Sketch y = f(x) + g(x) and y = f(x) − g(x) by adding or subtracting ordinates, including graphs with asymptotes, and find the domain of the result.

Adding ordinates

The ordinate of a point is its $y$-coordinate. To sketch $y = f(x) + g(x)$, add the two heights above each $x$-value. Some points are easy to place straight away:

Where $f(x) = 0$, the sum equals $g(x)$, so the sum meets the graph of $g$ there (and vice versa).
Where $f(x) = -g(x)$ (the graphs are mirror images in the $x$-axis), the sum is zero.
Where one graph has a vertical asymptote and the other is defined, the sum has the same asymptote.
Where $g(x) \to 0$ as $x \to \pm\infty$, the sum gets closer and closer to $y = f(x)$, which becomes an asymptote (possibly a slanted one).

y = x + k/x as a sum of two graphs

Orange: the sum. Blue dashed: y = x. Pink dashed: y = k/x. Far from the origin the pink graph is almost zero, so the orange curve hugs the line.

Try this

With $k = 1$, check that at $x = 1$ the ordinates $1$ and $1$ add to give the minimum value $2$. Then make $k$ negative.

Example 1●●

Sketch $y = x + \dfrac{1}{x}$.

Domain $x \ne 0$, with a vertical asymptote $x = 0$ inherited from $y = \dfrac1x$.

The curve approaches the line $y = x$ as an oblique asymptote.

1/x → 0 as x → ±∞.

Minimum turning point $(1, 2)$ for $x > 0$.

For x > 0, x + 1/x − 2 = (x − 1)²/x ≥ 0.

By symmetry, a maximum turning point $(-1, -2)$ for $x < 0$.

Both x and 1/x are odd, so the sum is odd.

Subtracting ordinates

For $y = f(x) - g(x)$, subtract the heights. The difference is zero where the two graphs intersect, positive where $f$ is above $g$, and negative where it is below. Another way to see it: $f - g = f + (-g)$, so reflect $g$ in the $x$-axis and add.

Example 2●●

Sketch $y = x - \sqrt{x}$.

Domain $x \ge 0$.

√x is only defined for x ≥ 0.

Zeroes at $x = 0$ and $x = 1$.

y = x and y = √x meet at x = 0 and x = 1.

Negative for $0 < x < 1$, positive for $x > 1$.

Between 0 and 1, √x is above x.

For large $x$, $x$ grows much faster than $\sqrt{x}$, so the graph rises steeply.

Flashcards

Ordinate

The $y$-coordinate of a point.

Where $f(x) = 0$

The sum $f + g$ equals $g(x)$.

Zeroes of $f - g$

Where the graphs of $f$ and $g$ intersect.

Domain of $f + g$

Where both $f$ and $g$ are defined.

Practice

$f(2) = 3$ and $g(2) = -1$. Find the ordinate of $y = f(x) + g(x)$ at $x = 2$.
$f(1) = 4$ and $g(1) = 6$. Find the ordinate of $y = f(x) - g(x)$ at $x = 1$.
The domain of $y = f(x) + g(x)$ is:
As $x \to \infty$, the graph of $y = x + \dfrac{1}{x}$ approaches:
Find the minimum value of $y = x + \dfrac{1}{x}$ for $x > 0$.
The zeroes of $y = f(x) - g(x)$ are where:
How many zeroes does $y = x - \sqrt{x}$ have?
The function $y = x + \dfrac{1}{x}$ is:
$y = x^2 + \dfrac{1}{x - 3}$ has one vertical asymptote. Enter its $x$-value.
Where $f(x) = 0$, the graph of $y = f(x) + g(x)$: