Question 508·Hard·Algebra

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The graph of $y = 2f(2x-1) - 2x + 1$ is shown. Which equation defines function $f$ ?

A function transformation question can show you the graph of a rule built from $f$ while asking for $f$ itself. Use labeled points in the entire graphed equation, not as points on $f$ . Because the graph is a line, write $f(x)=ax+b$ and use a Desmos element-list regression to find its slope $a$ and intercept $b$ .

Hints

Hint 1
Function notation means the whole expression $2x-1$ goes into $f$ . At the labeled point whose plotted $x$ -coordinate is $-2$ , what input does $f$ receive?
Hint 2
A plotted point gives an $x$ and a $y$ for the transformed graph. Put both coordinates into the graphed equation; don't assume the point gives you $f(-2)=5$ .
Hint 3
A linear function has the form $f(x)=ax+b$ , where $a$ is its slope and $b$ is its output at zero. How can the two labeled points give Desmos two conditions for $a$ and $b$ ?

Step-by-step

Fit the original function from the transformed points
Step 1
Separate the input from the output change
Let $u=2x-1$ , the whole input to $f$ . The remaining term, $-2x+1$ , equals $-u$ , so the graphed equation becomes $y=2f(u)-u$ . A point on the transformed graph is not automatically a point on $f$ .
Step 2
Use point P to get the first condition
At P, the plotted coordinates are $x=-2$ and $y=5$ . Type $2(-2)-1$ ; Desmos shows $-5$ , the input to $f$ . Putting $u=-5$ and $y=5$ into the rewritten equation gives $5=2f(-5)+5$ . The plotted $5$ is not $f(-2)$ .
Step 3
Use point Q to get the second condition
At Q, $x=6$ and $y=-1$ . Type $2(6)-1$ ; Desmos shows $11$ , so Q gives a condition on $f(11)$ : $-1=2f(11)-11$ . Keep the graphed output $-1$ separate from the function's output.
Step 4
Fit the line that satisfies both conditions
The graph is a line, and the input and output changes are linear, so write $f(x)=ax+b$ . Type that definition, then type $[2f(-5)+5,2f(11)-11]\sim[5,-1]$ . The $\sim$ asks Desmos to fit $a$ and $b$ to both plotted points. Under PARAMETERS it shows $a=0.3125$ and $b=1.5625$ . Type $a$ and $b$ on separate lines and use their fraction buttons to get $\frac{5}{16}$ and $\frac{25}{16}$ . So $f(x)=\frac{5}{16}x+\frac{25}{16}$ . Choice A.

Question 508·Hard·Algebra

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