Question

5 which point lies on the graph of $y = 3x^2 - \frac{1}{4}x + 3$?
(1) $(-2, 15.5)$
(2) $(-1, 5.75)$
(3) $(1, 6.25)$
(4) $(2, 15.5)$

6 given $f(x) = x^2$ and $g(x) = 8x - 15$ graphed on the same set of axes, which value(s) of $x$ will make $f(x) = g(x)$?
(1) 3, only
(2) 9, only
(3) 3 and 5
(4) 9 and 25

7 which trinomial is written in standard form and has a constant term of five?
(1) $x^5 - 4x^2 + 10$
(2) $2x^2 + 6x^4 + 5$
(3) $5x^4 - 3x^2 + 1$
(4) $4x^5 - 8x^2 + 5$

8 when solving $x^2 + 6x = -8$ for $x$, a student wrote $x^2 + 6x + 8 = 0$ as their first step. which property justifies this step?
(1) associative property
(2) commutative property
(3) zero property of addition
(4) addition property of equality

Explanation:

Question 5

Step1: Test point (-2,15.5)

Substitute \( x = -2 \) into \( y = 3x^2 - \frac{1}{4}x + 3 \).
\( y = 3(-2)^2 - \frac{1}{4}(-2) + 3 = 3(4) + 0.5 + 3 = 12 + 0.5 + 3 = 15.5 \).
This matches the y - value, but we check other points too.

Step2: Test point (-1,5.75)

Substitute \( x = -1 \).
\( y = 3(-1)^2 - \frac{1}{4}(-1) + 3 = 3(1) + 0.25 + 3 = 3 + 0.25 + 3 = 6.25
eq 5.75 \).

Step3: Test point (1,6.25)

Substitute \( x = 1 \).
\( y = 3(1)^2 - \frac{1}{4}(1) + 3 = 3 - 0.25 + 3 = 5.75
eq 6.25 \).

Step4: Test point (2,15.5)

Substitute \( x = 2 \).
\( y = 3(2)^2 - \frac{1}{4}(2) + 3 = 12 - 0.5 + 3 = 14.5
eq 15.5 \).
Only (-2,15.5) satisfies the equation.

Step1: Set \( f(x)=g(x) \)

Given \( f(x)=x^2 \) and \( g(x)=8x - 15 \), set \( x^2=8x - 15 \).

Step2: Rearrange to quadratic equation

\( x^2-8x + 15 = 0 \).

Step3: Factor the quadratic

\( (x - 3)(x - 5)=0 \).

Step4: Solve for x

Set each factor to zero: \( x - 3 = 0 \) or \( x - 5 = 0 \), so \( x = 3 \) or \( x = 5 \).

Step1: Recall standard form of a polynomial

Standard form of a polynomial is \( a_nx^n+\dots+a_1x + a_0 \), where exponents decrease.

Step2: Check constant term (term without x)

We need constant term 5.

Step3: Check each option

• Option (1): Constant term 10, not 5.
• Option (2): Not in standard form (exponents don't decrease: \( 2x^2+6x^4 + 5 \) should be \( 6x^4+2x^2 + 5 \)).
• Option (3): Constant term 1, not 5.
• Option (4): In standard form (\( 4x^5-8x^2 + 5 \), exponents decrease) and constant term 5.

Answer:

(1) (-2, 15.5)

Question 6