The graphs of
$f(x) = x^2 - 2x - 3$ and $g(x) = mx + c$ are drawn below - NSC Mathematics - Question 6 - 2024 - Paper 1
Question 6
The graphs of
$f(x) = x^2 - 2x - 3$ and $g(x) = mx + c$ are drawn below. D and E are the x-intercepts and P is the y-intercept of f. The turning point of f is T... show full transcript
Worked Solution & Example Answer:The graphs of
$f(x) = x^2 - 2x - 3$ and $g(x) = mx + c$ are drawn below - NSC Mathematics - Question 6 - 2024 - Paper 1
Step 1
6.1 Write down the range of f.
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Answer
To find the range of the function f(x)=x2−2x−3, we need to determine its vertex. The vertex form of a quadratic can be determined using the formula:
xv=−2ab=−2⋅1−2=1.
Substituting x=1 into f(x) gives:
f(1)=12−2(1)−3=1−2−3=−4.
As this is a parabola opening upwards, the range is [−4;∞).
Step 2
6.2 Calculate the coordinates of D and E.
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Answer
To find the x-intercepts (D and E), we set f(x)=0:
x2−2x−3=0.
Factoring gives:
(x−3)(x+1)=0.
Thus, the roots are x=3 and x=−1. Therefore, the coordinates are:
D(3; 0)
E(-1; 0)
Step 3
6.3 Determine the equation of g.
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Answer
Since P is the y-intercept of f, substituting x=0 gives:
f(0)=−3.
Assuming g(x)=mx+c with c=−3, the equation can be expressed as:
g(x)=mx−3.
Step 4
6.4 Write down the values of x for which f(x) - g(x) > 0.
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Answer
To find when f(x)−g(x)>0, we set up the inequality:
\Rightarrow x^2 - (2 - m)x > 0.$$
The solution will depend on the specific value of $m$ chosen.
Step 5
6.5 Determine the maximum vertical distance between h and g if h(x) = -f(x) for x ∈ [-2; 3].
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Answer
To find the vertical distance between h(x) and g(x), we evaluate:
d(x)=∣h(x)−g(x)∣=∣−f(x)−g(x)∣.
Calculating d(x) in the interval [−2;3]: substitute the bounds into the equation to determine the maximum distance.
Step 6
6.6 Given: k(x) = g(x) - n. Determine n if k is a tangent to f.
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Answer
For k(x) to be tangent to f(x), the discriminant must be zero:
D=b2−4ac=0.
This leads to finding n based on substituting g(x)−n into a form of f(x). The solution will yield n=2.25.
