Question 66·200 Super-Hard SAT Math Questions·Advanced Math

0 / 200 Questions completed

The graph of the quadratic function $y=f(x)$ intersects the $x$ -axis at $x=112$ and $x=k$ , where $k$ is a constant.

The graph also passes through the points $(18,47)$ and $(64,48)$ .

What is the value of $k$ ?

An $x$ -intercept gives you a point with output $0$ . Add that point to the other two, then use a Desmos quadratic regression to fit the graph and click its other intercept. Nearly equal outputs can be tempting to treat as mirror images, but symmetry requires the outputs to be exactly equal.

Hints

Hint 1
An $x$ -intercept is where the graph meets the horizontal axis, so its output is $0$ . Turn the known intercept into a full point. What three points can you enter in a Desmos table?
Hint 2
A quadratic regression fits a curve of the form $ax^2+bx+c$ to points. Put each input in the $x_1$ column and its matching output in the $y_1$ column.
Hint 3
The letters from a Desmos regression store the full fitted values, even when the displayed decimals are rounded. Define the function using those letters, then click the intercept other than the given one.

Step-by-step

Approach 1: Fit the three points in Desmos
Step 1
Make the intercept a point
An $x$ -intercept is where the graph meets the horizontal axis. So the intercept at $x=112$ gives the point $(112,0)$ . The output is $0$ , not $112$ .
Step 2
Fit a quadratic through the three points
Enter the three points in the table, with inputs in $x_1$ and outputs in $y_1$ . Type $y_1\sim ax_1^2+bx_1+c$ ; the $\sim$ tells Desmos to fit a quadratic, a function of the form $ax^2+bx+c$ . Under PARAMETERS, it reports $a\approx-0.01087$ , $b\approx0.91304$ , and $c\approx34.08696$ .
Step 3
Read the other intercept
Type $f(x)=ax^2+bx+c$ to graph the function using Desmos's stored coefficients, not rounded copies. Click the other $x$ -intercept at $(-28,0)$ . Its $x$ -coordinate is $k$ , so $k=-28$ . Choice A.

Approach 2: Use the known root as a factor
Step 1
Write the factored form
A root is an input where the output is $0$ . The roots $112$ and $k$ give $f(x)=a(x-112)(x-k)$ , where $a$ sets the parabola's scale. Each factor becomes zero at its root, so keep the factor signs opposite the roots.
Step 2
Use both given points
A point $(u,v)$ on the graph means $f(u)=v$ . Put each input into the same function and set it equal to its output: $47=a(18-112)(18-k)$ $48=a(64-112)(64-k)$ .
Step 3
Cancel the shared scale
Divide the first equation by the second to cancel $a$ : $\frac{47}{48}=\frac{(18-112)(18-k)}{(64-112)(64-k)}$ . This works because both points belong to one quadratic, so they have the same $a$ .
Step 4
Simplify the ratio
Subtract $112$ from each given input. Both differences are negative, so their signs cancel: $\frac{47}{48}=\frac{94(18-k)}{48(64-k)}$ . The only unknown left is $k$ .
Step 5
Clear the denominators
Multiply by $48(64-k)$ to get a linear equation: $47(64-k)=94(18-k)$ . We can do this because $k$ can't be $64$ : the given point $(64,48)$ has output $48$ , not $0$ .
Step 6
Solve for the missing root
Type $47(64-k)\sim94(18-k)$ . The $\sim$ tells Desmos to find the value of $k$ that makes both sides equal. Under PARAMETERS, it reports $k=-28$ , the other $x$ -intercept. Choice A.

Question 66·200 Super-Hard SAT Math Questions·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this