Functions

Try this

Move only $h$ . With $h = 2$ the formula contains $x - 2$ , yet the graph moves $2$ to the right. Why does a minus sign mean “right”? Look at the coordinates of $P$ and $P^{'}$ .
Press “Mirror in y-axis”. What has $x$ been replaced by in the formula? Then press “Mirror in x-axis”. What appears in front of the whole function value?
Switch to $F = ∣ f (x) ∣$ and to $F = f (∣ x ∣)$ and compare the two dashed graphs: one folds up and down, the other copies left and right. Which one is always an even function?
Switch on the level line $y = c$ and drag $c$ from bottom to top. When does the “number of solutions” change?

A mapping $f$ from $A$ to $B$ is called a function when $A$ and $B$ are non-empty sets of numbers. We write $y = f (x)$ , $x \in D$ . $D$ is the domain, and the set of values $f (D) = {f (x) ∣ x \in D}$ is the range. A function is determined by two things: its domain and its rule of correspondence.

Draw every pair $(x, f (x))$ as a point in the plane: together these points form the graph of the function. So “transforming a graph” just means moving each point on its own:

the point (a, b) lies on y = F (x) ⟺ the point (h + μ a, σ b + k) lies on y = σ F (μ (x - h)) + k .

Here $μ$ and $σ$ are $1$ or $- 1$ . Checking takes one line: substitute $x = h + μ a$ on the right; then $μ (x - h) = μ^{2} a = a$ , so $y = σ F (a) + k = σ b + k$ .

Predict first

How can the graph of $y = f (- x + 1)$ be obtained from the graph of $y = f (- x)$ by a shift?

shift left by

1

shift right by

1

shift up by

1

it depends on

f

Choose an answer first; then you can see the result.

Why: move it one point at a time

Example: from y = √x to y = 1 − √(2 − x)

Step 1 / 5

Follow one point
Take the point $P (1, 1)$ on $y = x$ . After each transformation, look where it has been moved to.

$y = x, P (1, 1)$
Mirror left–right: replace x by −x
Shift right by 2: replace x by x − 2
Mirror up–down: the value changes sign
Shift up by 1

Let the graph count the solutions

The number of real solutions of $g (x) = c$ is the number of points where the level line $y = c$ meets the graph $y = g (x)$ . For a function with an absolute value, fold the graph first and the answer is right in front of you.

Example: how many real solutions does |x² − 2x − 3| = c have?

Step 1 / 4

First draw |f(x)|
$f (x) = x^{2} - 2 x - 3 = (x - 1)^{2} - 4$ , with zeros $- 1$ and $3$ . Fold the part below the $x$ -axis ( $- 1 < x < 3$ ) up: a little “hill” appears in the middle, with its top at $(1, 4)$ .

$y = (x - 1)^{2} - 4$
c < 0 and c = 0
0 < c < 4
c = 4 and c > 4

Practice

Exercise 1Basic

Shift the graph of $y = x^{2}$ right by $2$ units and then up by $3$ units. Write down the function of the new graph (expanding it to the form $x^{2} + b x + c$ is fine too).

Exercise 2Basic

Reflect the graph of $y = x$ in the $y$ -axis, then shift it right by $3$ units. Call the resulting function $y = g (x)$ . Find $g (- 1)$ .

Exercise 3Advanced

How many real solutions does $∣ x^{2} - 2 x - 3∣ = 3$ have?

Exercise 4Challenge

$f (x) = x^{2} - 2 x - 3$ . If the equation $f (∣ x ∣) = c$ in $x$ has exactly $3$ real solutions, find $c$ .

Try this

Why: move it one point at a time

Example: from y = √x to y = 1 − √(2 − x)

Let the graph count the solutions

Example: how many real solutions does |x² − 2x − 3| = c have?

Practice

Mistake Museum · Exhibits from this stop

Seeing $+1$ and shifting right

Treating $|f(x)|$ and $f(|x|)$ as the same thing