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:
$$ f : x
ightarrow (x + 2)^2 - 4 $$
$$ g : x
ightarrow (x - 3... 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.
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To match the graphs:
For f: We evaluate the function at the important points using:
f(0)=(0+2)2−4=0
Thus, the graph intersects the y-axis at (0, 0).
For g: We check:
g(3)=(3−3)2−4=−4
Hence, g touches the y-value of -4 at x=3.
From these evaluations, we can determine that f corresponds to the graph on the left and g corresponds to the dashed graph on the right.
Step 2
Write down the roots of $f$ and the roots of $g$.
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The roots of f can be derived from the equation:
(x+2)2−4=0
Thus, solving gives:
(x+2)2=4x+2=ext±2
This results in: x=0,−4.
For the roots of g, we solve:
(x−3)2−4=0
Which simplifies to:
(x−3)2=4
Thus:
x−3=ext±2
Giving us: x=1,5.
Step 3
Sketch the graph of $h : x
ightarrow (x - 1)^2 - 4$ on the co-ordinate grid above.
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The graph of h can be sketched by recognizing that it follows a similar form to f and g. The vertex can be calculated:
(1,−4).
This vertex indicates the lowest point on the graph, which opens upwards. Adding symmetric points around the vertex will allow for a complete graph sketch.
Step 4
$p$ is a natural number, such that $(x - p)^2 - 2 = x^2 - 10x + 23$. Find the value of $p$.
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
First, we simplify the right-hand side:
(x−p)2−2=x2−10x+23
Rearranging gives:
(x−p)2=x2−10x+25
Thus, we find:
(x−p)2=(x−5)2
This implies that:
p=5, making p a natural number.
Step 5
Write down the equation of the axis of symmetry of the graph of the function $k(x) = x^2 - 10x + 23$.
97%
117 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The axis of symmetry for a quadratic function given by k(x)=ax2+bx+c can be computed using the formula:
x = - rac{b}{2a}.
For our function:
Here, a=1 and b=−10.
Therefore:
x = - rac{-10}{2 * 1} = 5.
Thus, the equation of the axis of symmetry is:
x=5.
Join the Junior Cycle students using SimpleStudy...
