Sure, let's break down the solution to these questions step-by-step.
2.1 What is the gradient of the given formula for calculating the costs?
For the given cost function [tex]\( y = 2x + 1 \)[/tex], the gradient (or slope) is the coefficient of [tex]\( x \)[/tex].
- Hence, the gradient is [tex]\( 2 \)[/tex].
2.2 When drawing a graph to represent the cost function, will the graph be increasing or decreasing? Provide a reason for your answer.
A graph is increasing if its gradient (slope) is positive and decreasing if the gradient is negative.
- Since the gradient is [tex]\( 2 \)[/tex], which is positive, the graph will be increasing. This means as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases.
2.3 Verify your answer in 2.2, by sketching the cost function using the dual-intercept method. Clearly indicate your intercepts.
To sketch the graph using the dual-intercept method, we need to determine the intercepts with the axes:
- Y-intercept: The y-intercept is where the graph crosses the y-axis (i.e., [tex]\( x = 0 \)[/tex]).
- Substituting [tex]\( x = 0 \)[/tex] into the cost function [tex]\( y = 2x + 1 \)[/tex]:
[tex]\[
y = 2(0) + 1 = 1
\][/tex]
- Thus, the y-intercept is [tex]\( (0, 1) \)[/tex].
- X-intercept: The x-intercept is where the graph crosses the x-axis (i.e., [tex]\( y = 0 \)[/tex]).
- To find the x-intercept, set [tex]\( y = 0 \)[/tex] in the cost function [tex]\( y = 2x + 1 \)[/tex]:
[tex]\[
0 = 2x + 1 \implies 2x = -1 \implies x = -\frac{1}{2}
\][/tex]
- Thus, the x-intercept is [tex]\( \left( -\frac{1}{2}, 0 \right) \)[/tex].
By plotting these intercepts, [tex]\( (0, 1) \)[/tex] and [tex]\( \left( -\frac{1}{2}, 0 \right) \)[/tex], we can see that the line connects the points indicating that the graph is indeed increasing, which matches our reasoning in 2.2.
