Question

create two additional quadratic functions, g(x) and h(x).
a. the function g(x) will open the same directions as f(x), have the same vertex, but will be narrower.
b. the function h(x) will open in the opposite direction as f(x), have the same vertex, but wo;; ne wider.

f(x)=2x^2-4x+30

please also explain because this chapter is extremely confusing

Answer 1

g(x) = 999999999999999999x^2-4x+30
h(x) = -.000000000000000001x^2-4x+30

Answer 2

(a)
the width of a quadratic is determined by the coefficient of the leading term. The larger the value the narrower the curve.

so the curves

\[f(x) = x^2\]

and

\[g(x) = 5x^2\]

both have their vertex at the origin, and the 2nd curve

\[ g(x) = 5x^2\]

is the narrower.

and easy way to check is with a table of values

x: -3 | -2 | -1 | 0 | 1 | 2 | 3
---------------------------------------------
f(x) | 9 | 4 | 1 | 0 | 1 | 4 | 9
-----------------------------------------------
g(x)| 45 | 20 | 5 | 0 | 5 | 20 | 45


if you plotted the points you would find g(x) is the narrower curve.




looking at (b)

you need to find the the vertex of the parabola

\[y = 2x^2 - 4x + 30\]


you need to write the curve in vertex form, first factor out the 2 and create a perfect square

\[y = a(x - h)^2 + k\]

where (h, k) is the vertex

the sign of a determines the concavity, positive concave up, negative concave down

in your question it can be written as

\[y = 2(x^2 -2x + 1) + 28

which can be written as

\[y = 2(x -1)^2 + 28\]

the vertex is (1, 28)

so the change the concavity

change the sign of 2

so the concave down curve with the same vertex is

\[y = -2(x -1)^2 + 28\]

you can distribute and collect like terms to get the equation in standard form

hope this helps