Write an equation for the following: 1. A square root graph with a locator point at (3,4) 2. An absolute value function shifted up 3 and vertically compressed

Answer 1

Prob. 1. A square root graph can be considered as the top of a parabola that's lying horizontally on its side. The locator point is the vertex of the parabola, in this case at (3,4).

Use the vertex form of a horizontal parabola (axis of symmetry is horizontal).

x = (y - k)² + h

So x = (y - 4)² + 3 would give the complete parabola, with 2 values of y for every 1 of x. To get this in the form of a square root equation, with y = some f(x) involving sq rt, move around the numbers with algebra.

x - 3 = (y - 4)²

^{Now take the square root of both sides, giving}

^{y - 4 = sq rt (x - 3)}

^{y = 4 + sq rt (x - 3)}

^{Prob. 2. There are many examples of an absolute value function shifted up 3 and vertically compressed.}

^{An absolute value function shifted up 3 could be simply y = | x | + 3.}

^{To compress it vertically, use a coefficient between 0 and 1 before the | | sign. This compresses it, as the vertical distance shrinks at each point. So an example could be y = 0.5 | x | + 3}

Answer 2

For any function, we can move the function around by adding constants inside and outside of the function. For a function f(x), we shift the function h units horizontally and k units vertically by changing the function f(x) to f(x-h)+k. "-h" is inside the function and "+k" is outside the function.

To shift the square root function, we are changing it from y = √x to y = √(x-h) + k. Note that the parenthesis tell us that "-h" is inside the square root and the "+k" is outside the square root. The function √x has a locator point at (0,0). So if we want to change our locator point to (3,4), we are going 3 units to the right and 4 units up. In other words, we plug in 3 for h and 4 for k.

So, the function should be y = √(x-3) + 4. Remember the parentheses tell us that "x-3" is inside the square root and "+4" is outside the square root.

Now, we can look at the second function, we want to shift vertically 3 units up, so we are going to add 3 outside the function. The absolute value function is y = |x|, so we change it to y = |x| + 3.

We can vertically compress (flatten) a function by adding any number between 0 and 1 as a coefficient. Coefficients go in front of the function. I will choose the coefficient 0.5 because it is between 0 and 1.

Adding the coefficient will give us the function y = 0.5|x| + 3.

Answer 3

Part 1:

y= √(mx+b)

y = (mx +b)^½

Guess: m=1

Given: (3,4)

x = 3; y = 4

plug in numbers and solve for b:

4 = (3 + b)^½

4^2 = 3 + b

16 = 3 + b

-3 -3

b = 13

Re-write the generic equation

y = (x + 13)^ ½

Part 2:

y=m |x| + b

b= 3 which will move the graph up 3 units

m = 1/2 or any fraction will flatten out the graph.

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