write a square root equation that has shifted down 7, right 4, reflected over the y axis, and horizontally stretched by 2. Then write the domain and range in interval notation.

Question

write a square root equation that has shifted down 7, right 4, reflected over the y axis, and horizontally stretched by 2. Then write the domain and range in interval notation.

Answer 1

To solve any problem asking to modify an equation, here are some general rules:

To shift an equation up or down, add or subtract from the equation outside of the main function.

For example, y=x^2 to shift this down 4 we subtract 4 from outside of the squared x function to look like y=((x)^2)-4

To shift an equation left or right, add or subtract from the equation inside of the main function.

For example, y=x^2 to shift this left 6 we add 6 from outside of the squared x function to look like y=((x+6)^2)

To flip an equation over the an axis, switch the sign in front of the other variable. (To flip over the x-axis, switch the y sign)

For example, y=((x+6)^2)+4 to flip over the y-axis, switch the x sign to look like y=((-x+6)^2)+4

To stretch an equation over x or over y, we multiply the main function by a constant. This constant will vary on which axis we are stretching, we multiply by a fraction, to make it longer over the x axis, and a whole number to make it longer over the y axis.

For example, y=((x+6)^2)+4 to stretch this over the y-axis by 8, (Make it taller by a factor of 8), multiply the (main)equation by 8 to look like y=(8(-x+6)^2)+4

Now, for your problem, we shift down by 7, right by 4, flip over y axis, and stretch it out over the x by a factor of 2. In that order...

y=SQRT(x) *Original*

y=SQRT(x)-7

y=SQRT(x-4)-7

y=SQRT(-x-4)-7

y=(1/2)SQRT(-x-4)-7

So your final answer is y=(1/2)SQRT(-x-4)-7

To figure out the domain and range of this function, we have to consider the original function, y=SQRT(x). The Domain, which is all x values included in the function, and the Range, which is all y values included in the function, of the original function is...

D: [0, INF) and R: [0, INF) Note that Infinity(INF) always has an open bracket.

Looking back at what we did to the original equation, we essentially moved that starting point, (0,0) to (-4,-7). We also changed the way x grows (Consider it as it now moves in the negative direction) but y is still increasing. So our new intervals for Domain and Range are...

D: (INF, -4] and R: [-7, INF)