Math

Question Find the equation of a square root function that passes through the points (0,2)(0,2) and (25,27)(25,27).

Studdy Solution

STEP 1

Assumptions
1. The function is a square root function.
2. The endpoint of the function is (0,2)(0,2).
3. The function passes through the point (25,27)(25,27).

STEP 2

A general form for a square root function is given by:
f(x)=axh+kf(x) = a\sqrt{x - h} + k
where (h,k)(h,k) is the endpoint of the function and aa is a scaling factor.

STEP 3

Since the endpoint is (0,2)(0,2), we can substitute h=0h = 0 and k=2k = 2 into the general form of the square root function.
f(x)=ax+2f(x) = a\sqrt{x} + 2

STEP 4

Now we need to determine the value of aa. We can use the point (25,27)(25,27) which lies on the function to find aa.

STEP 5

Substitute x=25x = 25 and f(x)=27f(x) = 27 into the equation from STEP_3.
27=a25+227 = a\sqrt{25} + 2

STEP 6

Since 25=5\sqrt{25} = 5, we can simplify the equation:
27=5a+227 = 5a + 2

STEP 7

Subtract 2 from both sides to isolate the term with aa:
272=5a27 - 2 = 5a
25=5a25 = 5a

STEP 8

Divide both sides by 5 to solve for aa:
255=a\frac{25}{5} = a
5=a5 = a

STEP 9

Now that we have determined a=5a = 5, we can write the final equation for the square root function.
f(x)=5x+2f(x) = 5\sqrt{x} + 2
This is the equation of the square root function with endpoint (0,2)(0,2) passing through the point (25,27)(25,27).

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