The quadratic function intersects the -axis at and and the -axis at . What is the value of ?
Two -intercepts cue a quadratic’s factored form: each zero supplies a factor. Use the -intercept to find the remaining multiplier with a one-unknown Desmos regression, then evaluate the function. The midpoint of the zeros locates the parabola’s symmetry line, but it isn’t the requested output. Keep the input and output separate.
Hints
- Hint 1
An -intercept has output . Each given intercept supplies a factor that becomes at its input. Leave a multiplier in front: knowing where a parabola crosses the axis doesn’t yet tell you its height elsewhere.
- Hint 2
The -intercept is the output at input . Use to make an equation for the unknown multiplier. Why can this point find it when either -intercept cannot?
- Hint 3
In function notation, asks for the output when the input is . The number lies halfway between the zeros, but being the middle input doesn’t make it the output.
Step-by-step
Build the factored rule, then evaluate
Step 1Turn the zeros into factors
At an -intercept, the output is . The factors and become at the two given inputs, so write . The roots give the factors; the -intercept finds their multiplier. Keep : without it, you’d assume the multiplier is .
- Step 2
Use the point that can find the multiplier
The -intercept means input gives output . Put those numbers into the rule: . Use this point rather than a root: at a root, the product is already , whatever is.
- Step 3
Find the multiplier in Desmos
Type . The regression symbol tells Desmos to find ; under PARAMETERS it shows . Type on the next line and use its fraction button to see .
- Step 4
Evaluate at the requested input
Type , then . Desmos uses the stored and shows ; the fraction button gives . This negative output fits the factored form: at , the factors are and . So . Choice B.
Lessons that teach this
- SAT Nonlinear FunctionsIntermediateCoreBuild a quadratic function from conditions
- SAT Nonlinear FunctionsAdvancedCoreUse the three forms of a quadratic function
- SAT Nonlinear FunctionsIntermediateEvaluate nonlinear functions and recover inputs
- DesmosAdvancedCoreFactor, root, and remainder tests
- DesmosBeginnerEvaluate, combine, and compose functions