Two -intercepts point to factored form: each intercept gives a factor that becomes zero there. The factors leave one multiplier unknown, so use the -intercept to find it with a one-unknown Desmos regression. Then evaluate the requested input. The midpoint between the intercepts locates the axis of symmetry, but you still need the function's value there.
Hints
- Hint 1
An -intercept has an output of zero. For each intercept, write a factor that becomes zero at its -value, then put an unknown multiplier in front of the two factors.
- Hint 2
A -intercept gives the output when the input is . Substitute that input into your factored rule and set it equal to the given output. What number must the multiplier be?
- Hint 3
Once you know the multiplier, define the function in Desmos and enter . The number inside the parentheses is the input, not a new intercept.
Step-by-step
Build the factored rule, then evaluate
Step 1Turn the intercepts into factors
An -intercept is where the output is zero. The factors and become zero at the two given inputs, so the quadratic has factored form . The multiplier is unknown because the intercepts alone don't fix the curve's height.
- Step 2
Turn the other intercept into an equation
The -intercept means input gives output , or . Put into the rule: . Unlike the roots, this point gives a nonzero product, so the extra intercept fixes the missing multiplier.
- Step 3
Find the multiplier in Desmos
Type . The tilde tells Desmos to find the multiplier that fits the equation. Under PARAMETERS, Desmos shows .
- Step 4
Evaluate the function at 3
Type , then . Desmos uses the fitted multiplier and shows . The midpoint is not a root: at , neither factor is zero. So the function's output at is . Choice B.