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. b) The line y = mx cuts the graph of y = f(x) in two distinct places. Find all possible values of m.

SSCE HSC Mathematics Advanced Absolute value functions: The graph of y = f(x), where f(x

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.

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Answer

To solve for a, b, and c, we will use the given points.

From the point (6, 7):

We can substitute into the function:

$f(6) = a|6 - b| + c = 7$
From the point (3, -5):

$f(3) = a|3 - b| + c = -5$
From the point (9, -5):

$f(9) = a|9 - b| + c = -5$

Given that the graph is shifted 7 units vertically upwards, we can conclude:

Therefore, c = 7.
Knowing the graph shifts to the right by 6 units gives us b = 6.

We can now substitute these values back into one of the equations to find 'a'. Choosing the second equation:

$a|3 - 6| + 7 = -5$
This simplifies to:
$3a + 7 = -5 \\ 3a = -12 \\ a = -4$

Thus, the values are:

a = -4
b = 6
c = 7.

Step 2

Find all possible values of m.

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Answer

To find the values of m such that the line y = mx intersects the graph of y = f(x) at two distinct points:

The slope of the line through points (6, 7) and (0, 0) is:

$m = \frac{7 - 0}{6 - 0} = \frac{7}{6}$
For the line to intersect the graph twice, we need:
- The slope m must be less than $\frac{7}{6}$
- Additionally, since it needs to intersect below the vertex (at around (6, 7)), m must be greater than -4 to ensure it cuts the graph above the x-axis.

Combining these conditions, we get:

$-4 < m < \frac{7}{6}$

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

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

Find the values of a, b and c.

Find all possible values of m.

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