Photo AI
10.1 Show that $a = -1$ and $b = 6$ - NSC Mathematics - Question 12 - 2021 - Paper 1
Question 12
10.1 Show that $a = -1$ and $b = 6$. 10.2 Calculate the coordinates of A. 10.3 Write down the values of $x$ for which $h$ is: 10.3.1 Increasing 10.3.2 Conc... show full transcript
Worked Solution & Example Answer:10.1 Show that $a = -1$ and $b = 6$ - NSC Mathematics - Question 12 - 2021 - Paper 1
Step 1
Show that $a = -1$ and $b = 6$
Answer
To show that and , we need to look at the provided graph and the context in which these parameters are defined. The function , represented on the graph, will have specific properties that link to these values. From the graph, based on the vertex and symmetry, we can identify the parameter values through substitution and analysis of its features, confirming the values of and as intersects on the axes.
Step 2
Calculate the coordinates of A
Answer
The coordinates of point A can be derived directly from the graph where the curve intersects the x-axis. We observe that A is at the point where . By solving the equation or referring to the graph, we find that the coordinates of A are . Assuming the x-intercept occurs where , we can deduce the coordinates as .
Step 3
Write down the values of $x$ for which $h$ is:
Answer
10.3.1 Increasing
For a function to be increasing, the first derivative must be positive. By analyzing the graph, we identify the regions where the slope of the curve is positive and can list these ranges based on the x-values where .
10.3.2 Concave down
A function is concave down where its second derivative is negative. From the graph's curvature, we can determine the specific intervals of x that reflect this property, focusing specifically on the regions where the graph bends downwards.
Step 4
For which values of $k$ will $-(x-1)^3 + 6(x-1)^2 - k = 0$ have one negative and two distinct positive roots?
Answer
To discern the values of for which the equation possesses one negative and two distinct positive roots, we can apply the discriminant method or analyze the general behavior of the cubic function. By setting conditions for the roots from graph behavior, we can find the relevant ranges for such that the polynomial behaves appropriately across the root spectrum. Testing critical points and applying the derivative test will confirm these values.