Question 817·Medium·Math

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The function $h$ is defined for all real numbers $x$ by an equation of the form

h(x)=k(0.5)^{x}+4,

where $k$ is a constant. If the point $(2,6)$ lies on the graph of $h$ , which of the following equations could represent $h$ ?

An exponential function with one unknown coefficient and a point on its graph gives you an equation for that coefficient. Put the point’s input and output into the given rule, then use a Desmos regression to find the missing number. Don’t stop at checking the point: different formulas can give the same output for one input.

Hints

Hint 1
A point $(a,b)$ on a function’s graph means that input $a$ gives output $b$ . For the given point, what goes in place of $x$ , and what goes in place of $h(x)$ ?
Hint 2
The coefficient $k$ multiplies the exponential part, so it isn’t necessarily the difference between the given output and the fixed term. What value of $k$ makes your input-output equation true?

Step-by-step

Use the point to find the coefficient
Step 1
Turn the point into an equation
A point on a function’s graph gives an input-output equation. The point $(2,6)$ says $h(2)=6$ , so replace $x$ with $2$ and $h(x)$ with $6$ in the given rule:
$6=k(0.5)^2+4.$
Step 2
Find the coefficient in Desmos
Type $6\sim k(0.5)^2+4$ . The tilde tells Desmos to find the value of $k$ that makes the equation true. Under PARAMETERS, Desmos shows $k=8$ .
Step 3
Put the coefficient into the required form
Substitute $k=8$ into the original rule: $h(x)=8(0.5)^x+4$ . Keep the $+4$ : passing through the point alone isn’t enough if an equation changes the given form. This equation represents $h$ . Choice C.

Question 817·Medium·Math

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Step-by-step

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Strategy

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Step-by-step

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