7 the function ( f(x) ) is shown in the table below.
\begin{array}{|c|c|c|c|c|c|c|c|}hlinex & 0 & 3 & 2 & 6 & 1 & 5 & 4 & m \hlinef(x) & 6 & 2 & 7 & 5 & 8 & 4 & 3 & 9 \hlineend{array}
state an appropriate value for ( m ) in the table, so that ( f(x) ) remains a function. explain your reasoning.

8 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

9 the sum of ( (x + 7)^2 ) and ( (x - 3)^2 ) is
1) ( 2x^2 + 58 )
2) ( 2x^2 + 58 )
3) ( 2x^2 + 8x + 58 )
4) ( 2x^4 + 8x^2 + 58 )

12 the axis of symmetry is ( x = 2 ) for which quadratic function?
(there is a graph of a quadratic function and a table for ( g(x) ) with ( x = -2, -1, 0 ) and ( g(x) = 6, 3, 2 ))

Question

7 the function ( f(x) ) is shown in the table below.
\begin{array}{|c|c|c|c|c|c|c|c|}hlinex & 0 & 3 & 2 & 6 & 1 & 5 & 4 & m \hlinef(x) & 6 & 2 & 7 & 5 & 8 & 4 & 3 & 9 \hlineend{array}
state an appropriate value for ( m ) in the table, so that ( f(x) ) remains a function. explain your reasoning.

8 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

9 the sum of ( (x + 7)^2 ) and ( (x - 3)^2 ) is
1) ( 2x^2 + 58 )
2) ( 2x^2 + 58 )
3) ( 2x^2 + 8x + 58 )
4) ( 2x^4 + 8x^2 + 58 )

12 the axis of symmetry is ( x = 2 ) for which quadratic function?
(there is a graph of a quadratic function and a table for ( g(x) ) with ( x = -2, -1, 0 ) and ( g(x) = 6, 3, 2 ))

Accepted Answer

### Question 7
# Explanation:
## Step1: Recall function definition
A function is a relation where each input $ x $ has exactly one output $ f(x) $. So, for $ f(x) $ to be a function, the new $ x $-value $ m $ must not be equal to any existing $ x $-values in the table.
## Step2: Identify existing $ x $-values
The existing $ x $-values in the table are $ 0, 3, 2, 6, 1, 5, 4 $.
## Step3: Choose a valid $ m $
We can choose any value for $ m $ that is not in $ \{0, 3, 2, 6, 1, 5, 4\} $. For example, let's choose $ m = 7 $ (any number not in the existing set works, like 8, 9, -1, etc.).

# Answer:
An appropriate value for $ m $ is $ 7 $ (or any number not in $ \{0, 3, 2, 6, 1, 5, 4\} $). Reason: A function requires each $ x $-value to have exactly one $ f(x) $-value, so $ m $ must not duplicate any existing $ x $-value.

### Question 8
# Explanation:
## 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 $ gives $ x = 3 $; $ x - 5 = 0 $ gives $ x = 5 $.

# Answer:
3 and 5 (Option 3: 3 and 5)

### Question 9
# Explanation:
## Step1: Expand $ (x + 7)^{2} $
Using the formula $ (a + b)^{2}=a^{2}+2ab + b^{2} $, we get $ (x + 7)^{2}=x^{2}+14x + 49 $.
## Step2: Expand $ (x - 3)^{2} $
Using the formula $ (a - b)^{2}=a^{2}-2ab + b^{2} $, we get $ (x - 3)^{2}=x^{2}-6x + 9 $.
## Step3: Sum the two expanded forms
$ (x^{2}+14x + 49)+(x^{2}-6x + 9)=x^{2}+14x + 49 + x^{2}-6x + 9 $.
## Step4: Combine like terms
Combine $ x^{2} $ terms: $ x^{2}+x^{2}=2x^{2} $; combine $ x $ terms: $ 14x-6x = 8x $; combine constant terms: $ 49 + 9 = 58 $. So the sum is $ 2x^{2}+8x + 58 $.

# Answer:
$ 2x^{2}+8x + 58 $ (Option 3)

### Question 12 (Assuming we need to find the function with axis of symmetry $ x = 2 $)
For a quadratic function $ f(x)=ax^{2}+bx + c $, the axis of symmetry is $ x=-\frac{b}{2a} $. Let's check the given function $ f(x)=2x^{2}+8x $:
## Step1: Identify $ a $ and $ b $
Here, $ a = 2 $, $ b = 8 $.
## Step2: Calculate axis of symmetry
$ x=-\frac{8}{2\times2}=-\frac{8}{4}=-2 $. Wait, maybe the other function? Wait, the table for $ g(x) $: let's find the quadratic for $ g(x) $. The points are $ (-2,6) $, $ (-1,3) $, $ (0,2) $. Let's assume $ g(x)=ax^{2}+bx + c $. For $ x = 0 $, $ c = 2 $. For $ x=-1 $: $ a(-1)^{2}+b(-1)+2 = 3\Rightarrow a - b = 1 $. For $ x=-2 $: $ a(-2)^{2}+b(-2)+2 = 6\Rightarrow 4a - 2b = 4\Rightarrow 2a - b = 2 $. Subtract $ a - b = 1 $ from $ 2a - b = 2 $: $ (2a - b)-(a - b)=2 - 1\Rightarrow a = 1 $, then $ b = 0 $. So $ g(x)=x^{2}+2 $, axis of symmetry $ x = 0 $. Wait, maybe the first function's graph: the parabola has axis of symmetry $ x = 2 $. Let's check the function $ f(x)=2x^{2}+8x $: no, as above. Wait, maybe a typo, but if we consider the graph with axis of symmetry $ x = 2 $, the function with $ a = 2 $, $ b=-8 $ (since $ x = -\frac{b}{2a}=2\Rightarrow -\frac{b}{4}=2\Rightarrow b=-8 $), so $ f(x)=2x^{2}-8x + c $. But maybe the question is to identify the function with axis of symmetry $ x = 2 $. Assuming the first function's graph (the parabola) has axis of symmetry $ x = 2 $, so the function is the one with vertex at $ x = 2 $. But based on the given $ f(x)=2x^{2}+8x $, no. Wait, maybe the question is which function (between the two) has axis of symmetry $ x = 2 $. But with the given info, let's re - check. Wait, maybe the original problem's function: if we have a parabola opening upwards with vertex at $ x = 2 $, then the function with axis of symmetry $ x = 2 $ is the one whose graph is shown (the parabola), and the function $ f(x)=2x^{2}+8x $ has axis of symmetry $ x=-2 $, so maybe the graph's function (the parabola) has axis of symmetry $ x = 2 $, so the answer is the function whose graph is the parabola (the first one, labeled 1) with axis of symmetry $ x = 2 $. But maybe the question is to confirm, but based on the options, if we assume the function with axis of symmetry $ x = 2 $ is the parabola in the graph (the first function), so the answer is the function whose graph is the parabola (option 1's graph). But maybe the question is incomplete, but if we proceed with the given, the function with axis of symmetry $ x = 2 $ is the quadratic function whose graph is shown (the parabola), and among the given functions, if we calculate for $ f(x)=2x^{2}+8x $, no, but maybe a mistake. However, if we take the graph's function, the axis of symmetry is $ x = 2 $, so the function is the one with the parabola (option 1's graph).

# Answer:
The function with axis of symmetry $ x = 2 $ is the parabola shown in the graph (the first function, labeled 1). (Assuming the graph's function has axis of symmetry $ x = 2 $)

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QUESTION IMAGE

Question

Explanation:

Question 7

Step1: Recall function definition

Step2: Identify existing \( x \)-values

Step3: Choose a valid \( m \)

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

Step2: Rearrange to quadratic equation

Step3: Factor the quadratic

Step4: Solve for \( x \)

Step1: Expand \( (x + 7)^{2} \)

Step2: Expand \( (x - 3)^{2} \)

Step3: Sum the two expanded forms

Step4: Combine like terms

Answer:

Question 8