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The graphs of two functions, f and g, are shown on the co-ordinate grid below - Junior Cycle Mathematics - Question 11 - 2014
Question 11
The graphs of two functions, f and g, are shown on the co-ordinate grid below. The functions are: f : x ↦ (x + 2)² - 4 g : x ↦ (x - 3)³ - 4. (a) Match the graph... show full transcript
Worked Solution & Example Answer:The graphs of two functions, f and g, are shown on the co-ordinate grid below - Junior Cycle Mathematics - Question 11 - 2014
Step 1
Match the graphs to the functions by writing f or g beside the corresponding graph on the grid.
Answer
To determine which graph corresponds to which function, we evaluate the functions:
-
For function f:
f(0) = (0 + 2)² - 4 = 4 - 4 = 0.
-
For function g:
g(0) = (0 - 3)³ - 4 = -27 - 4 = -31.
Based on these calculations, graph f is the parabola that opens upwards and has its vertex at (-2, -4), while graph g is a cubic function with a point of inflection around (3, -4). Thus, we plot "f" next to f and "g" next to g on the grid.
Step 2
Write down the roots of f and the roots of g.
Answer
The roots of f can be found by setting the function equal to zero:
- (x + 2)² - 4 = 0
This implies:
- (x + 2)² = 4
- x + 2 = ±2
Therefore, x = 0 and x = -4 are the roots of f.
For g, we likewise set it to zero:
- (x - 3)³ - 4 = 0
This gives:
- (x - 3)³ = 4
- x - 3 = ± oot{3}{4}
So, the roots of g are x = 3 + oot{3}{4} and x = 3 - oot{3}{4}.
Step 3
Sketch the graph of h : x ↦ (x - 1)² - 4 on the co-ordinate grid above, where x ∈ R.
Answer
To sketch the graph of h(x), we recognize that it is a parabola that shifts the vertex of f downwards. The vertex is at (1, -4). We evaluate:
- h(-1) = ((-1) - 1)² - 4 = 0 - 4 = -4
- h(1) = (0) - 4 = -4
- h(3) = (2)² - 4 = 0
Using these points and the symmetry of the parabola, we can sketch the graph above.
Step 4
p is a natural number, such that (x - p)² - 2 = x² - 10x + 23.
Answer
Rearranging the equation, we have:
(x - p)² - 2 = x² - 10x + 23
Now factoring this, we get:
(x - p)² = x² - 10x + 25.
This implies:
(x - p) = ±(x - 5)
To satisfy this equality, we must have p = 5. Therefore, the value of p is 5.
Step 5