This is in response to Andrew's question posted earlier

Question

If f(x)=5x^2,. Find the new equation for a minimum of 7 at x=14. While maintaining the same shape as the graph F(x)=5x^2.

This was asked by Andrew

If you do a horizontal translation of the graph and a vertical translation of the graph that would be equivalent to shifting the graph horizontally and vertically.

Horizontal translation are when you add the number to x inside the parentheses:

F(x+14)=5(x+14)^2

F(x+14)=5(x+14)(x+14)

Foil gives:

x^2+14x+14x + 196

5*(x^2+ 28x +196)

5x^2+ 140x + 980

This is for F(x+14)

Then do a vertical translation of the graph:

Vertical translation are reflected by adding the number to the end or subtracting it to the end.

We know that our vertex point because we are given information about the minimum value is at 7 while at the x value of 14. So this would have had to have been shifted up 7 on the y-axis and shifted to the right by 14 on the x-axis.

F(x)=5x^2 +7

Would now reflect this vertical translation.

If you combine both functions you would have:

F(x) shifted vertically +7 + F(x+14)

Combine the terms for both and you have:

(5x^2+7) + (5x^2 +140x+ 980)

When combining like terms you get:

10x^2 +140x +987

Notice that when adding 5x^2 to 5x^2 it becomes 10x^2, not 10x^4. This is because of the power rule. If you add the same bases but when multiplying for example x^2 * x^2, then it becomes x^4. When adding x^2+ x^2 it just becomes 2x^2.

So the new F(x) function would be:

F(x)= 10x^2 + 140x +987

This would be the new equation, the shape would be the same because the highest polynomial order or leading term is still x^2.

The shape would still be quadratic or parabola. The number out front in front of x determines how vertically compressed it is while the highest leading polynomial ie x^2, x^3, x^4 determines shape.

Answer 1

f(x) = 5x²

Since the minimum is at (14, 7) the vertex is at (14, 7)

g(x) = 5(x - 14)² + 7