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The graph of y = f(x), where f(x) = a|x - b| + c, passes through the points (3, –5), (6, 7) and (9, –5) as shown in the diagram - HSC - SSCE Mathematics Advanced - Question 27 - 2023 - Paper 1
Question 27
The graph of y = f(x), where f(x) = a|x - b| + c, passes through the points (3, –5), (6, 7) and (9, –5) as shown in the diagram. (a) Find the values of a, b and c. ... show full transcript
Worked Solution & Example Answer:The graph of y = f(x), where f(x) = a|x - b| + c, passes through the points (3, –5), (6, 7) and (9, –5) as shown in the diagram - HSC - SSCE Mathematics Advanced - Question 27 - 2023 - Paper 1
Step 1
Find the values of a, b and c
Answer
To determine the values of a, b, and c, we start by noting that the vertex of the absolute value function represents the point at which the function reaches its maximum value.
Given the vertex (6, 7), we know:
- The graph is shifted vertically upwards by 7 units. Hence, c = 7.
- The graph is also shifted 6 units to the right, so b = 6.
Now, let's use the point (3, -5) to find a.
From the equation f(x) = a|x - b| + c:
-
Plugging in (3, -5):
f(3) = a|3 - 6| + 7 = -5
|3 - 6| = 3, so we have:-5 = 3a + 7
3a = -12
a = -4.
Thus, the values are
- a = -4,
- b = 6,
- c = 7.
Step 2
Find all possible values of m
Answer
For the line y = mx to intersect the graph twice, we analyze the slope condition.
The slope of the line joining (6, 7) to (0, 0) is given by:
slope = \frac{7 - 0}{6 - 0} = \frac{7}{6}.
To have two intersection points, the line must have a slope less than \frac{7}{6}. Therefore:
- m < \frac{7}{6}.
Additionally, we consider the right segment of the graph of f(x) which has a negative slope.
To intersect this segment, the slope must be greater than -4. So we establish:
-4 < m < \frac{7}{6}.
Hence, the values of m can be represented as:
-4 < m < \frac{7}{6}.