part ii: quadratic relations
part a: multiple choice: circle the correct answer. 7 marks
1. the graph of ( y = x^2 - x - 12 ) will cross the ( y )-axis at
a) 1
b) (-1)
c) (-12)
d) 12
2. the expanded form of ( (x + 4)(x - 2) ) is
a) ( 2x^2 + 4x - 8 )
b) ( x^2 + 2x - 8 )
c) ( 2x^2 - 8 )
d) ( x^2 - 2x - 8 )
3. the factored form of ( x^2 - 2x - 15 ) is
a) ( (x + 5)(x - 3) )
b) ( (x - 5)(x + 3) )
c) ( (x - 5)(x + 10) )
d) ( (x - 5)(x - 3) )
4. when the equation ( (x - 3)(x + 8) = 0 ) is solved, the solutions, also called the zeroes, are
a) (-3) and 8
b) (-24) and 0
c) 5 and 0
d) 3 and (-8)
5. a parabola with vertex ((-4, 0)) has the equation of the axis of symmetry
a) ( x = 0 )
b) ( y = 0 )
c) ( x = -4 )
d) ( y = -4 )
6. which of the following statements is false regarding the parabola defined by the relation ( y = x^2 )?
a) it represents a parabola in standard position.
b) the vertex of the parabola lies at the origin.
c) the axis of symmetry is the ( y )-axis.
d) the point ( (2, -4) ) lies on the curve.
7. what is the factored form of the expression ( -21x^2 + 72 )?
a) ( -3(7x^2 - 24) )
b) ( 3(7x^2 - 24) )
c) ( -3(7x^2 + 24) )
d) ( 3(7x^2 + 24) )

Question

part ii: quadratic relations
part a: multiple choice: circle the correct answer. 7 marks
1. the graph of ( y = x^2 - x - 12 ) will cross the ( y )-axis at
a) 1
 b) (-1)
 c) (-12)
 d) 12
2. the expanded form of ( (x + 4)(x - 2) ) is
a) ( 2x^2 + 4x - 8 )
 b) ( x^2 + 2x - 8 )
 c) ( 2x^2 - 8 )
 d) ( x^2 - 2x - 8 )
3. the factored form of ( x^2 - 2x - 15 ) is
a) ( (x + 5)(x - 3) )
 b) ( (x - 5)(x + 3) )
 c) ( (x - 5)(x + 10) )
 d) ( (x - 5)(x - 3) )
4. when the equation ( (x - 3)(x + 8) = 0 ) is solved, the solutions, also called the zeroes, are
a) (-3) and 8
 b) (-24) and 0
 c) 5 and 0
 d) 3 and (-8)
5. a parabola with vertex ((-4, 0)) has the equation of the axis of symmetry
a) ( x = 0 )
 b) ( y = 0 )
 c) ( x = -4 )
 d) ( y = -4 )
6. which of the following statements is false regarding the parabola defined by the relation ( y = x^2 )?
a) it represents a parabola in standard position.
 b) the vertex of the parabola lies at the origin.
 c) the axis of symmetry is the ( y )-axis.
 d) the point ( (2, -4) ) lies on the curve.
7. what is the factored form of the expression ( -21x^2 + 72 )?
a) ( -3(7x^2 - 24) )
 b) ( 3(7x^2 - 24) )
 c) ( -3(7x^2 + 24) )
 d) ( 3(7x^2 + 24) )

Accepted Answer

### Question 1
# Explanation:
## Step1: Recall y - axis crossing rule
To find where a graph crosses the y - axis, we set $x = 0$ in the equation of the function.
## Step2: Substitute $x = 0$ into $y=x^{2}-x - 12$
When $x = 0$, we have $y=0^{2}-0 - 12=- 12$.
# Answer: c) $-12$

### Question 2
# Explanation:
## Step1: Use the distributive property (FOIL method)
The formula for $(a + b)(c + d)=ac+ad+bc+bd$. For $(x + 4)(x - 2)$, we have $a=x$, $b = 4$, $c=x$, $d=-2$.
So, $(x + 4)(x - 2)=x\times x+x\times(-2)+4\times x + 4\times(-2)$
## Step2: Simplify the terms
$x\times x=x^{2}$, $x\times(-2)=-2x$, $4\times x = 4x$, $4\times(-2)=-8$. Then combine like terms: $x^{2}-2x + 4x-8=x^{2}+2x - 8$
# Answer: b) $x^{2}+2x - 8$

### Question 3
# Explanation:
## Step1: Recall factoring of quadratic $x^{2}+bx + c=(x + m)(x + n)$ where $m + n=b$ and $mn=c$
For $x^{2}-2x - 15$, we need two numbers $m$ and $n$ such that $m + n=-2$ and $mn=-15$. The numbers are $-5$ and $3$ since $-5+3=-2$ and $-5\times3=-15$.
## Step2: Write the factored form
So, $x^{2}-2x - 15=(x - 5)(x + 3)$ (because $x-5=x+(-5)$ and $x + 3$)
# Answer: b) $(x - 5)(x + 3)$

### Question 4
# Explanation:
## Step1: Recall zero - product property
If $ab = 0$, then either $a = 0$ or $b = 0$. For $(x - 3)(x + 8)=0$, we set each factor equal to zero.
## Step2: Solve for $x$
Case 1: $x-3=0$, then $x = 3$. Case 2: $x + 8=0$, then $x=-8$.
# Answer: d) $3$ and $-8$

### Question 5
# Explanation:
## Step1: Recall the axis of symmetry of a parabola
For a parabola with vertex $(h,k)$, the equation of the axis of symmetry is $x = h$.
## Step2: Identify $h$ from the vertex $(-4,0)$
Here, $h=-4$ and $k = 0$. So the axis of symmetry is $x=-4$.
# Answer: c) $x=-4$

### Question 6
# Explanation:
## Step1: Analyze each option
- Option a: The parabola $y = x^{2}$ has its vertex at the origin and opens upward, so it is in standard position. This statement is true.
- Option b: The vertex of $y=x^{2}$ is at $(0,0)$ (the origin). This statement is true.
- Option c: For $y=x^{2}$, the axis of symmetry is the $y$ - axis ($x = 0$) since the parabola is symmetric about the $y$ - axis. This statement is true.
- Option d: Substitute $x = 2$ into $y=x^{2}$, we get $y=2^{2}=4
eq-4$. So the point $(2,-4)$ does not lie on the curve. This statement is false.
# Answer: d) the point $(2,-4)$ lies on the curve.

### Question 7
# Explanation:
## Step1: Factor out the greatest common factor (GCF)
First, find the GCF of $-21x^{2}$ and $72$. The GCF of $21$ and $72$ is $3$, and since the first term is negative, we factor out $-3$.
## Step2: Divide each term by $-3$
$-21x^{2}\div(-3) = 7x^{2}$ and $72\div(-3)=-24$. So, $-21x^{2}+72=-3(7x^{2}-24)$
# Answer: a) $-3(7x^{2}-24)$

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QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Recall y - axis crossing rule

Step2: Substitute \(x = 0\) into \(y=x^{2}-x - 12\)

Step1: Use the distributive property (FOIL method)

Step2: Simplify the terms

Step1: Recall factoring of quadratic \(x^{2}+bx + c=(x + m)(x + n)\) where \(m + n=b\) and \(mn=c\)

Step2: Write the factored form

Answer:

Question 2