Certainly! Let's go through each part of the question step-by-step.
### 7.1 Value of [tex]\( q \)[/tex]
Given that the line [tex]\( y = -6 \)[/tex] is an asymptote to the graph of [tex]\( h \)[/tex], we can directly determine that [tex]\( q = -6 \)[/tex].
### 7.2 Calculate the value of [tex]\( a \)[/tex]
We are given that the graph of [tex]\( h \)[/tex] passes through the point [tex]\(\left(-1, -5.25\right)\)[/tex]. Using the general form of the function [tex]\( h(x) = a \cdot 2^{x-1} + q \)[/tex]:
1. Substitute [tex]\( x = -1 \)[/tex], [tex]\( y = -5.25 \)[/tex], and [tex]\( q = -6 \)[/tex] into the equation:
[tex]\[
-5.25 = a \cdot 2^{-1-1} - 6
\][/tex]
2. Simplify the exponent:
[tex]\[
-5.25 = a \cdot 2^{-2} - 6
\][/tex]
3. Recognize that [tex]\( 2^{-2} = \frac{1}{4} \)[/tex]:
[tex]\[
-5.25 = a \cdot \frac{1}{4} - 6
\][/tex]
4. Rearrange the equation to solve for [tex]\( a \)[/tex]:
[tex]\[
-5.25 + 6 = a \cdot \frac{1}{4}
\][/tex]
5. Simplify:
[tex]\[
0.75 = \frac{a}{4}
\][/tex]
6. Multiply both sides by 4:
[tex]\[
a = 3
\][/tex]
### 7.3 Calculate the average gradient between the x-intercept and the y-intercept of [tex]\( h \)[/tex]
To find the y-intercept, evaluate [tex]\( h(0) \)[/tex]:
[tex]\[
h(0) = 3 \cdot 2^{0-1} - 6 = 3 \cdot \frac{1}{2} - 6 = 1.5 - 6 = -4.5
\][/tex]
Thus, the y-intercept is [tex]\( (0, -4.5) \)[/tex].
Next, to find the x-intercept, solve [tex]\( h(x) = 0 \)[/tex]:
[tex]\[
0 = 3 \cdot 2^{x-1} - 6
\][/tex]
1. Add 6 to both sides:
[tex]\[
6 = 3 \cdot 2^{x-1}
\][/tex]
2. Divide both sides by 3:
[tex]\[
2 = 2^{x-1}
\][/tex]
3. Since [tex]\( 2 = 2^1 \)[/tex], [tex]\( x-1 = 1 \)[/tex]:
[tex]\[
x = 2
\][/tex]
Thus, the x-intercept is [tex]\( (2, 0) \)[/tex].
The average gradient (slope) between the y-intercept [tex]\((0, -4.5)\)[/tex] and the x-intercept [tex]\((2, 0)\)[/tex] is:
[tex]\[
\text{Average gradient} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-4.5)}{2 - 0} = \frac{4.5}{2} = 2.25
\][/tex]
### 7.4 Determine the equation of [tex]\( p \)[/tex] if [tex]\( p(x) = h(x-2) \)[/tex]
To find [tex]\( p(x) \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( x-2 \)[/tex] in the original function [tex]\( h(x) \)[/tex]:
[tex]\[
p(x) = h(x-2) = 3 \cdot 2^{(x-2)-1} - 6
\][/tex]
Simplify the expression inside the exponent:
[tex]\[
p(x) = 3 \cdot 2^{x-3} - 6
\][/tex]
So, the equation of [tex]\( p \)[/tex] is:
[tex]\[
p(x) = 3 \cdot 2^{x-3} + (-6)
\][/tex]
### Summary
Here are the results summarized:
1. The value of [tex]\( q \)[/tex] is [tex]\(-6\)[/tex].
2. The value of [tex]\( a \)[/tex] is [tex]\( 3 \)[/tex].
3. The y-intercept value is [tex]\(-4.5\)[/tex], and the x-intercept value is [tex]\( 2 \)[/tex].
4. The average gradient between the x-intercept and the y-intercept is [tex]\( 2.25 \)[/tex].
5. The equation of [tex]\( p \)[/tex] is [tex]\( p(x) = 3 \cdot 2^{x-3} - 6 \)[/tex].
