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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 - 2011
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)² - 4. (a) Match the ... 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 - 2011
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 each function, we can substitute specific values into the functions and compare with the graphs.
Calculating for function f:
- For x = 0: f(0) = (0 + 2)² - 4 = 0.
Calculating for function g:
- For x = 0: g(0) = (0 - 3)² - 4 = 5.
Examining the graphs, we find that:
- The graph showing f is the one that intersects at (0, 0), hence it is labeled accordingly.
- The graph showing g is the dashed one that has a value of 5 at x = 0.
Step 2
Write down the roots of f and the roots of g .
Answer
To find the roots, we will set each function equal to zero:
-
Roots of f: (x + 2)² - 4 = 0
This simplifies to (x + 2)² = 4 leading us to: x + 2 = ±2, therefore the roots are x = 0 and x = -4.
-
Roots of g: (x - 3)² - 4 = 0
Similarly simplifying gives us: (x - 3)² = 4 leading to: x - 3 = ±2, therefore the roots are x = 1 and x = 5.
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 function h, we note that it is a transformation of the basic parabolic shape:
-
The vertex form gives the vertex at (1, -4) since (x - 1) indicates a horizontal shift of 1 unit to the right and the -4 indicates a downward shift of 4 units.
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The parabola opens upwards. By substituting a few values:
x h(x) 0 -3 1 -4 2 -3 3 -2
This shape should be plotted on the grid effectively as a symmetric shape about the line x = 1.
Step 4
p is a natural number, such that (x - p)² - 2 = -x² - 10x + 23. Find the value of p.
Answer
To solve for p , first rearrange the equation:
(x - p)² - 2 = -x² - 10x + 23
Adding 2 to both sides gives:
(x - p)² = -x² - 10x + 25.
Now, we need to observe that the right-hand side can be factored or rearranged:
Recognizing the formula, the equation can be expressed as:
(x - 5)² = (x - p)²
This means p = 5.
Step 5
Write down the equation of the axis of symmetry of the graph of the function:
Answer
Using part (d), we see that:
The equation of the axis of symmetry for a quadratic function in the form k(x) = ax² + bx + c is given by:
x = - rac{b}{2a}.
For the function k(x) = x² - 10x + 23, we get:
Here, a = 1 and b = -10, thus:
x = - rac{-10}{2 * 1} = 5.
Therefore, the axis of symmetry is x = 5.