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### 7.1 Write down the value of [tex]\( q \)[/tex].
The line [tex]\( y = -6 \)[/tex] is an asymptote to the graph of [tex]\( h(x) \)[/tex], and from the equation [tex]\( h(x) = a \cdot 2^{x-1} + q \)[/tex], it is clear that [tex]\( q \)[/tex] represents the horizontal asymptote.
Thus, the value of [tex]\( q \)[/tex] is:
[tex]\[ q = -6 \][/tex]
### 7.2 Calculate the value of [tex]\( a \)[/tex] if the graph passes through the point [tex]\((-1, -5.25)\)[/tex].
Given the point [tex]\((-1, -5.25)\)[/tex] passes through [tex]\( h(x) \)[/tex]:
[tex]\[ h(-1) = -5.25 \][/tex]
Substitute [tex]\( x = -1 \)[/tex] into the equation [tex]\( h(x) = a \cdot 2^{x-1} + q \)[/tex]:
[tex]\[ -5.25 = a \cdot 2^{-2} - 6 \][/tex]
[tex]\[ -5.25 = a \cdot \frac{1}{4} - 6 \][/tex]
[tex]\[ -5.25 + 6 = a \cdot \frac{1}{4} \][/tex]
[tex]\[ 0.75 = a \cdot \frac{1}{4} \][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[ a = 0.75 \cdot 4 \][/tex]
[tex]\[ a = 3.0 \][/tex]
### 7.3 Calculate the average gradient between the [tex]\( x \)[/tex]-intercept and the [tex]\( y \)[/tex]-intercept of [tex]\( h \)[/tex].
#### Finding the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = a \cdot 2^{0-1} + q \][/tex]
[tex]\[ h(0) = a \cdot 2^{-1} - 6 \][/tex]
[tex]\[ h(0) = 3.0 \cdot \frac{1}{2} - 6 \][/tex]
[tex]\[ h(0) = 1.5 - 6 \][/tex]
[tex]\[ h(0) = -4.5 \][/tex]
#### Finding the [tex]\( x \)[/tex]-intercept:
The [tex]\( x \)[/tex]-intercept occurs when [tex]\( h(x) = 0 \)[/tex]:
[tex]\[ 0 = a \cdot 2^{x-1} + q \][/tex]
[tex]\[ 0 = 3.0 \cdot 2^{x-1} - 6 \][/tex]
[tex]\[ 6 = 3.0 \cdot 2^{x-1} \][/tex]
[tex]\[ 2 = 2^{x-1} \][/tex]
[tex]\[ x-1 = \log_2(2) \][/tex]
[tex]\[ x-1 = 1 \][/tex]
[tex]\[ x = 2 \][/tex]
#### Calculating the average gradient:
The average gradient between the [tex]\( x \)[/tex]-intercept and the [tex]\( y \)[/tex]-intercept is the change in [tex]\( h \)[/tex] values over the change in [tex]\( x \)[/tex] values.
Change in [tex]\( h \)[/tex] values (from [tex]\( y \)[/tex]-intercept, which is [tex]\(-4.5\)[/tex], to [tex]\( h \)[/tex] value at [tex]\( y = 0 \)[/tex]):
[tex]\[ \Delta h = -4.5 - 0 = -4.5 \][/tex]
Change in [tex]\( x \)[/tex] values (from [tex]\( x \)[/tex]-intercept, which is 2, to [tex]\( x = 0 \)[/tex]):
[tex]\[ \Delta x = 0 - 2 = -2 \][/tex]
Average gradient:
[tex]\[ \text{Average gradient} = \frac{\Delta h}{\Delta x} = \frac{-4.5}{-2} = 2.25 \][/tex]
### 7.4 Determine the equation of [tex]\( p \)[/tex] if [tex]\( p(x) = h(x-2) \)[/tex] in the form [tex]\( p(x) = a \cdot 2^{x-1} + q \)[/tex].
Substitute [tex]\( x-2 \)[/tex] into the original function:
[tex]\[ p(x) = h(x-2) = a \cdot 2^{(x-2)-1} + q \][/tex]
[tex]\[ p(x) = 3.0 \cdot 2^{x-3} - 6 \][/tex]
Hence, the equation of [tex]\( p \)[/tex] is:
[tex]\[ p(x) = 3.0 \cdot 2^{x-3} - 6 \][/tex]
### Summary:
1. [tex]\( q = -6 \)[/tex]
2. [tex]\( a = 3.0 \)[/tex]
3. Average gradient = [tex]\( 2.25 \)[/tex]
4. [tex]\( p(x) = 3.0 \cdot 2^{x-3} - 6 \)[/tex]
### 7.1 Write down the value of [tex]\( q \)[/tex].
The line [tex]\( y = -6 \)[/tex] is an asymptote to the graph of [tex]\( h(x) \)[/tex], and from the equation [tex]\( h(x) = a \cdot 2^{x-1} + q \)[/tex], it is clear that [tex]\( q \)[/tex] represents the horizontal asymptote.
Thus, the value of [tex]\( q \)[/tex] is:
[tex]\[ q = -6 \][/tex]
### 7.2 Calculate the value of [tex]\( a \)[/tex] if the graph passes through the point [tex]\((-1, -5.25)\)[/tex].
Given the point [tex]\((-1, -5.25)\)[/tex] passes through [tex]\( h(x) \)[/tex]:
[tex]\[ h(-1) = -5.25 \][/tex]
Substitute [tex]\( x = -1 \)[/tex] into the equation [tex]\( h(x) = a \cdot 2^{x-1} + q \)[/tex]:
[tex]\[ -5.25 = a \cdot 2^{-2} - 6 \][/tex]
[tex]\[ -5.25 = a \cdot \frac{1}{4} - 6 \][/tex]
[tex]\[ -5.25 + 6 = a \cdot \frac{1}{4} \][/tex]
[tex]\[ 0.75 = a \cdot \frac{1}{4} \][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[ a = 0.75 \cdot 4 \][/tex]
[tex]\[ a = 3.0 \][/tex]
### 7.3 Calculate the average gradient between the [tex]\( x \)[/tex]-intercept and the [tex]\( y \)[/tex]-intercept of [tex]\( h \)[/tex].
#### Finding the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = a \cdot 2^{0-1} + q \][/tex]
[tex]\[ h(0) = a \cdot 2^{-1} - 6 \][/tex]
[tex]\[ h(0) = 3.0 \cdot \frac{1}{2} - 6 \][/tex]
[tex]\[ h(0) = 1.5 - 6 \][/tex]
[tex]\[ h(0) = -4.5 \][/tex]
#### Finding the [tex]\( x \)[/tex]-intercept:
The [tex]\( x \)[/tex]-intercept occurs when [tex]\( h(x) = 0 \)[/tex]:
[tex]\[ 0 = a \cdot 2^{x-1} + q \][/tex]
[tex]\[ 0 = 3.0 \cdot 2^{x-1} - 6 \][/tex]
[tex]\[ 6 = 3.0 \cdot 2^{x-1} \][/tex]
[tex]\[ 2 = 2^{x-1} \][/tex]
[tex]\[ x-1 = \log_2(2) \][/tex]
[tex]\[ x-1 = 1 \][/tex]
[tex]\[ x = 2 \][/tex]
#### Calculating the average gradient:
The average gradient between the [tex]\( x \)[/tex]-intercept and the [tex]\( y \)[/tex]-intercept is the change in [tex]\( h \)[/tex] values over the change in [tex]\( x \)[/tex] values.
Change in [tex]\( h \)[/tex] values (from [tex]\( y \)[/tex]-intercept, which is [tex]\(-4.5\)[/tex], to [tex]\( h \)[/tex] value at [tex]\( y = 0 \)[/tex]):
[tex]\[ \Delta h = -4.5 - 0 = -4.5 \][/tex]
Change in [tex]\( x \)[/tex] values (from [tex]\( x \)[/tex]-intercept, which is 2, to [tex]\( x = 0 \)[/tex]):
[tex]\[ \Delta x = 0 - 2 = -2 \][/tex]
Average gradient:
[tex]\[ \text{Average gradient} = \frac{\Delta h}{\Delta x} = \frac{-4.5}{-2} = 2.25 \][/tex]
### 7.4 Determine the equation of [tex]\( p \)[/tex] if [tex]\( p(x) = h(x-2) \)[/tex] in the form [tex]\( p(x) = a \cdot 2^{x-1} + q \)[/tex].
Substitute [tex]\( x-2 \)[/tex] into the original function:
[tex]\[ p(x) = h(x-2) = a \cdot 2^{(x-2)-1} + q \][/tex]
[tex]\[ p(x) = 3.0 \cdot 2^{x-3} - 6 \][/tex]
Hence, the equation of [tex]\( p \)[/tex] is:
[tex]\[ p(x) = 3.0 \cdot 2^{x-3} - 6 \][/tex]
### Summary:
1. [tex]\( q = -6 \)[/tex]
2. [tex]\( a = 3.0 \)[/tex]
3. Average gradient = [tex]\( 2.25 \)[/tex]
4. [tex]\( p(x) = 3.0 \cdot 2^{x-3} - 6 \)[/tex]
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