The functions $f(x) = -x^2 - 2x + 3$ and $g(x) = mx + c$ are drawn below, with $g$ passing through $E$, $C$ and $A$ - NSC Mathematics - Question 4 - 2016 - Paper 1
Question 4
The functions $f(x) = -x^2 - 2x + 3$ and $g(x) = mx + c$ are drawn below, with $g$ passing through $E$, $C$ and $A$. A and B are the x-intercepts of $f$, and $CD$ is... show full transcript
Worked Solution & Example Answer:The functions $f(x) = -x^2 - 2x + 3$ and $g(x) = mx + c$ are drawn below, with $g$ passing through $E$, $C$ and $A$ - NSC Mathematics - Question 4 - 2016 - Paper 1
Step 1
Determine the coordinates of C, the turning point of the graph of f.
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Answer
To find the turning point of the quadratic function f(x)=−x2−2x+3, we first need to use the vertex formula. The x-coordinate of the vertex (turning point) can be found using:
x=−2ab
where a=−1 and b=−2. Thus:
x=−2(−1)−2=1
Next, substitute x=1 back into the function to find the y-coordinate:
f(1)=−12−2(1)+3=0
Therefore, the coordinates of C are (1,0).
Step 2
Determine the coordinates of A and B.
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Answer
To find the x-intercepts (points A and B) of the graph f(x), set f(x)=0:
−x2−2x+3=0
Multiplying through by -1, we get:
x2+2x−3=0
Factoring the quadratic:
(x+3)(x−1)=0
Thus, the solutions are:
x=−3extandx=1
This means the coordinates of A and B are A(−3,0) and B(1,0). However, since point C is on the graph as well, only A has a distinct x-coordinate.
Step 3
Determine the values of m and c.
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Answer
Since line g(x)=mx+c passes through point C (1, 0) and point E (at the y-intercept of g), we can write two equations. The line passes through C, hence:
0=m(1)+c
Thus:
c=−m
Using the coordinates of another point known, we would need to know the coordinates of E to solve for m and c. If not known, we shall retain the equation c=−m.
Step 4
Calculate the length of CE. (leave your answer in surd form)
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Answer
To find length CE, we would use the Euclidean distance formula:
∣CE∣=(xE−1)2+(yE−0)2
If coordinates of point E are known, substitute them in; if E is at (xE,yE), we can resolve to get the length.
Step 5
Determine the values of x, for which f(x), g(x) < 0.
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Answer
To find the x-values for which both f(x) and g(x) are less than 0:
Analyze f(x)<0. Already established intercepts at A(-3, 0) and B(1, 0) give us the interval (−3,1) where f(x) is negative.
For g(x)<0, we need to set mx+c<0 and solve the linear inequality using known coordinates. Thus identify the overlapping intervals from both inequalities found.
