Question

which of the following is the graph of $f(x) = -3(x - 1)^2 + 12$?
a)

b)

c)

d)

1. simplify the expression $sqrt{18x} + 4sqrt{8x} - sqrt{50x}$.

a. $13sqrt{2x}$
b. $6sqrt{2x}$
c. $72x$
d. $left( sqrt{18} + 4sqrt{8} - sqrt{50}
ight)sqrt{x}$

1. simplify the expression $sqrt{45b} + 4sqrt{20b} - 2sqrt{125b}$.

a. $sqrt{5b}$
b. $12sqrt{5b}$
c. $58$
d. $left( sqrt{45} + 4sqrt{20} - 2sqrt{125}
ight)sqrt{b}$

Explanation:

Question 37

Step1: Simplify each radical

Simplify $\sqrt{18x}$: $\sqrt{9\times2x} = 3\sqrt{2x}$
Simplify $4\sqrt{8x}$: $4\sqrt{4\times2x} = 4\times2\sqrt{2x} = 8\sqrt{2x}$
Simplify $\sqrt{50x}$: $\sqrt{25\times2x} = 5\sqrt{2x}$

Step2: Combine like terms

Substitute back into the expression:
$3\sqrt{2x} + 8\sqrt{2x} - 5\sqrt{2x}$
Combine coefficients: $(3 + 8 - 5)\sqrt{2x} = 6\sqrt{2x}$

Step1: Simplify each radical

Simplify $\sqrt{45b}$: $\sqrt{9\times5b} = 3\sqrt{5b}$
Simplify $4\sqrt{20b}$: $4\sqrt{4\times5b} = 4\times2\sqrt{5b} = 8\sqrt{5b}$
Simplify $2\sqrt{125b}$: $2\sqrt{25\times5b} = 2\times5\sqrt{5b} = 10\sqrt{5b}$

Step2: Combine like terms

Substitute back into the expression:
$3\sqrt{5b} + 8\sqrt{5b} - 10\sqrt{5b}$
Combine coefficients: $(3 + 8 - 10)\sqrt{5b} = \sqrt{5b}$

The function is $f(x) = -3(x - 1)^2 + 12$, which is a quadratic in vertex form $f(x) = a(x - h)^2 + k$.

• Vertex: $(h, k) = (1, 12)$
• Direction: Since $a = -3 < 0$, the parabola opens downward.

Check the graphs:

• Option a: Vertex near (1, 11-12), opens downward.
• Option b: Vertex higher, not (1,12).
• Option c: Opens upward (incorrect direction).
• Option d: Vertex not at (1,12) and opens downward but vertex position off.

The correct graph should have vertex at (1,12) and open downward, matching option a (or rechecking, likely option b if vertex Y is 12? Wait, re-express:
Wait, $f(1) = 12$, so vertex at (1,12). Let’s check y-intercept: $f(0) = -3(1) + 12 = 9$. So y-intercept at (0,9).

Looking at the graphs:

• Graph a: y-intercept ~6-7? No.
• Graph b: y-intercept ~10-12? Wait, $f(0) = 9$, so y-intercept 9. Let’s recalculate: $f(0) = -3(0 - 1)^2 + 12 = -3 + 12 = 9$. So y-intercept (0,9).

Graph b: y-intercept around 9-10, vertex at (1,12), opens downward. So likely b? Wait, original graphs:

Wait, the function $f(x) = -3(x-1)^2 + 12$:

• Vertex (1,12)
• Opens downward
• Y-intercept at 9

Looking at the options:

• a: Vertex lower, y-intercept ~6
• b: Vertex at (1,12) (since grid: x=1, y=12), y-intercept ~9, opens downward.
• c: Opens upward (wrong)
• d: Vertex not at (1,12)

So correct graph is b.

Answer:

B. $6\sqrt{2x}$

Question 38