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:
$$f: x o (x + 2)^2 - 4$$
$$g: x o (x - 3)^2 - 4$$
(a) Match ... 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.
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Answer
To determine which graph corresponds to each function, we can evaluate their values at specific points:
For f:
f(0)=(0+2)2−4=0
For g:
g(0)=(0−3)2−4=5
From the evaluations, we note that the graph of f will intersect the y-axis at 0 and is the parabola opening upwards on the grid, thus it must be on the left. Therefore, we designate;
Graph corresponding to f: Left parabola
Graph corresponding to g: Right dashed section.
Step 2
Write down the roots of $f$ and the roots of $g$.
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Answer
To find the roots, we set each function equal to zero:
For f:
(x+2)2−4=0
Solving gives:
(x+2)2=4x+2=ext±2
Thus, the roots are:
x=0,−4
For g:
(x−3)2−4=0
Solving gives:
(x−3)2=4x−3=ext±2
Thus, the roots are:
x=1,5
Step 3
Sketch the graph of $h: x \to (x - 1)^2 - 4$ on the co-ordinate grid above, where $x \in \mathbb{R}$.
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Answer
To sketch the graph of h, calculate several key points:
h(−1)=(−1−1)2−4=0−4=−4
h(1)=(1−1)2−4=0−4=−4
h(3)=(3−1)2−4=4−4=0
Plotting these points on the grid will show a parabola opening upwards with its vertex at (1,−4) and symmetry about x=1.
Step 4
Find the value of $p$.
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Answer
To solve for p in the equation (x−p)2−2=x2−10x+23, we first rearrange it:
(x−p)2=x2−10x+25
Analyzing the right side reveals:
(x−5)2
Thus, we equate:
x−p=x−5
So, we find that:
p=5.
Step 5
Write down the equation of the axis of symmetry of the graph of the function: $k(x) = x^2 - 10x + 23$.
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Answer
The axis of symmetry for a quadratic function in the form k(x)=ax2+bx+c can be found using the formula:
x=−2ab
Here, a=1 and b=−10, thus:
x=−2(1)−10=5
Therefore, the equation of the axis of symmetry is:
x=5.
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