Question

released items
published october 2018
nc math 1
north carolina
end-of-course
assessment
nc math 1—released items
1 which choice is the graph of ( y = (8 - x)(x + 2) )?
(options a, b, c, d with graphs)
nc math 1—released items
2 in which graph does the shaded region represent the solution set for the inequality shown below?
( 2x - y < 4 )
(options a, b, c, d with graphs)
nc math 1—released items
3 which expression is equivalent to ( (x + 2)(3x - 3) )?
a ( 3x^2 - 6 )
b ( 3x^2 + 3x - 6 )
c ( 3x^2 + 6x - 6 )
d ( 3x^2 + 9x - 6 )

Explanation:

Question 3 Solution:

Step 1: Apply the distributive property (FOIL method)

To expand \((x + 2)(3x - 2)\), we use the distributive property (also known as the FOIL method for binomials). The FOIL method stands for First, Outer, Inner, Last.

First: Multiply the first terms in each binomial: \(x \times 3x = 3x^2\)

Outer: Multiply the outer terms: \(x \times (-2) = -2x\)

Inner: Multiply the inner terms: \(2 \times 3x = 6x\)

Last: Multiply the last terms: \(2 \times (-2) = -4\) Wait, no, wait the second binomial is \(3x - 2\)? Wait, no, the original problem is \((x + 2)(3x - 2)\)? Wait, no, looking back, the problem is \((x + 2)(3x - 3)\)? Wait, the user's image shows "Which expression is equivalent to \((x + 2)(3x - 3)\)?" Let's correct that.

So, \((x + 2)(3x - 3)\)

First: \(x \times 3x = 3x^2\)

Outer: \(x \times (-3) = -3x\)

Inner: \(2 \times 3x = 6x\)

Last: \(2 \times (-3) = -6\)

Now, combine like terms:

\(3x^2 + (-3x + 6x) - 6 = 3x^2 + 3x - 6\) Wait, but let's check again. Wait, maybe I misread the problem. Wait, the options are:

A. \(3x^2 - 6\)

B. \(3x^2 + 3x - 6\)

C. \(3x^2 + 6x - 6\)

D. \(3x^2 + 9x - 6\)

Wait, let's do the expansion correctly.

\((x + 2)(3x - 3)\)

Using the distributive property (also called the distributive law or the FOIL method for binomials):

\(x \times 3x = 3x^2\)

\(x \times (-3) = -3x\)

\(2 \times 3x = 6x\)

\(2 \times (-3) = -6\)

Now, combine the middle terms: \(-3x + 6x = 3x\)

So the expanded form is \(3x^2 + 3x - 6\), which is option B.

Wait, but let's check again. Wait, maybe the problem was \((x + 2)(3x - 2)\), but the image shows \((x + 2)(3x - 3)\). Let's confirm.

Alternatively, maybe a typo, but based on the options, let's proceed.

So, step by step:

1. Expand \((x + 2)(3x - 3)\) using the distributive property (FOIL):

\(= x(3x - 3) + 2(3x - 3)\)

\(= 3x^2 - 3x + 6x - 6\)

1. Combine like terms:

\(-3x + 6x = 3x\), so we have \(3x^2 + 3x - 6\)

To determine which graph represents the solution set for the inequality \(2x - y < 4\), we can follow these steps:

Step 1: Rewrite the inequality in slope - intercept form (\(y=mx + b\))

Start with \(2x - y < 4\).
Subtract \(2x\) from both sides: \(-y < - 2x+4\).
Multiply both sides by \(- 1\). Remember that when we multiply or divide an inequality by a negative number, the direction of the inequality sign changes. So we get \(y>2x - 4\).

Step 2: Analyze the boundary line

The boundary line for the inequality \(y > 2x-4\) has a slope \(m = 2\) and a \(y\) - intercept \(b=-4\). Since the inequality is \(y>2x - 4\) (not \(y\geq2x - 4\)), the boundary line should be a dashed line (to indicate that the points on the line are not included in the solution set).

Step 3: Determine the region to shade

To find which side of the line to shade, we can use a test point. A common test point is \((0,0)\) (as long as it is not on the boundary line).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>2x - 4\):
\(0>2(0)-4\)
\(0>-4\), which is a true statement. So we shade the region that contains the point \((0,0)\).

Now let's analyze the graphs:

• Graph A: If the line is not dashed or the shading is incorrect, we can eliminate it.
• Graph B: Let's check the boundary line and shading. The boundary line should have a slope of 2 and \(y\) - intercept of - 4. If the shading is on the side that includes \((0,0)\) and the line is dashed, this could be a candidate.
• Graph C: If the line or the shading direction is wrong, we can eliminate it.
• Graph D: If the line or the shading direction is wrong, we can eliminate it.

After analyzing the slope, the type of line (dashed), and the shading direction, we find that the graph that represents \(y>2x - 4\) (the solution to \(2x - y < 4\)) is the one where the boundary line is dashed, has a slope of 2 and \(y\) - intercept of - 4, and the shading is on the side containing \((0,0)\). From the given graphs, the correct graph is the one that matches these criteria (assuming the graphs are labeled such that the correct one has a dashed line with slope 2, \(y\) - intercept - 4, and shading above the line).

Question 1 Solution:

To determine which graph is the graph of \(y=(8 - x)(x + 2)\), we can follow these steps:

Step 1: Find the x - intercepts

Set \(y = 0\), then \((8 - x)(x + 2)=0\).
Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
So, \(8 - x=0\) gives \(x = 8\) and \(x + 2=0\) gives \(x=-2\). So the x - intercepts of the parabola are at \(x=-2\) and \(x = 8\).

Step 2: Find the y - intercept

Set \(x = 0\) in the equation \(y=(8 - x)(x + 2)\). Then \(y=(8-0)(0 + 2)=8\times2 = 16\). So the y - intercept is at \((0,16)\).

Step 3: Determine the direction of the parabola

First, expand the equation \(y=(8 - x)(x + 2)=8x+16-x^{2}-2x=-x^{2}+6x + 16\).
For a quadratic equation of the form \(y = ax^{2}+bx + c\), if \(a<0\), the parabola opens downwards. Here, \(a=-1<0\), so the parabola opens downwards.

Now, let's analyze the graphs:

• The parabola should have x - intercepts at \(x=-2\) and \(x = 8\), a y - intercept at \((0,16)\), and open downwards. We look for the graph that has these characteristics (x - intercepts at - 2 and 8, y - intercept at 16, and opening downwards).

Final Answers:

• For Question 1: The graph with x - intercepts at \(x=-2\) and \(x = 8\), y - intercept at \((0,16)\) and opening downwards.
• For Question 2: The graph with a dashed line \(y = 2x-4\) and shading above the line (containing \((0,0)\)).
• For Question 3: B.…

Answer:

B. \(3x^2 + 3x - 6\)

Question 2 Solution: