The graphs of two functions, $f$ and $g$, are shown on the co-ordinate grid below. The functions are: $f : x \mapsto (x + 2)^2 - 4$ g : $x \mapsto (x - 3)^2 - 4$. (a) Match the graphs to the functions by writing $f$ or $g$ beside the corresponding graph on the grid. (b) Write down the roots of $f$ and the roots of $g$. Roots of $f$: Roots of $g$: (c) Sketch the graph of $h : x \mapsto (x - 1)^2 - 4$ on the co-ordinate grid above, where $x \in \mathbb{R}$. d) $p$ is a natural number, such that $(x - p)^2 = 2x^{-1} - 10x + 23$. Find the value of $p$. e) Write down the equation of the axis of symmetry of the graph of the function: $k(x) = x^2 - 10x + 23$.

Junior Cycle Mathematics Functions: The graphs of two functions, $f$

The graphs of two functions, $f$ and $g$, are shown on the co-ordinate grid below - Junior Cycle Mathematics - Question 11 - 2015

Question 11

The graphs of two functions, $f$ and $g$, are shown on the co-ordinate grid below. The functions are: $f : x \mapsto (x + 2)^2 - 4$ g : $x \mapsto (x - 3)^2 - 4$. ... 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 - 2015

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 the functions, we can evaluate the functions:

For $f(0)$ :

$f(0) = (0 + 2)^2 - 4 = 0$ ,
For $g(0)$ :

$g(0) = (0 - 3)^2 - 4 = 5$ .

This indicates that the graph of $f$ does intersect the x-axis and is the one on the left, while the graph of $g$ is on the right.

Step 2

Write down the roots of $f$ and the roots of $g$.

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Answer

The roots of $f$ can be found by setting the function equal to zero:

$(x + 2)^2 - 4 = 0$
We solve this to find $x + 2 = ightarrow 0$ or $x + 2 = ±2$ . Thus, the roots are $x = -2$ and $x = 0$ .

For $g$ , we also set the function to zero:

$(x - 3)^2 - 4 = 0$
From this, we get $x - 3 = ±2$ . So the roots are $x = 1$ and $x = 5$ .

Step 3

Sketch the graph of $h: x \mapsto (x - 1)^2 - 4$ on the co-ordinate grid above, where $x \in \mathbb{R}$.

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Answer

To sketch the graph of $h$ , we recognize that it is a parabola that opens upwards with its vertex at the point (1,-4). The values of $h$ can be calculated for several values around 1 (e.g., at $x = 0$ , $x = 2$ , and $x = 1$ ) to provide shape. Plotting these points will give an accurate representation.

Step 4

Find the value of $p$.

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Answer

To solve for $p$ , we start from the equation:

$(x - p)^2 = 2x^{-1} - 10x + 23$
Rearranging this, we notice that we need to isolate the equation:

$(x - p)^2 = -10x + 23$
Identifying that $p$ is a natural number, we can examine suitable values for $p$ (given relations to the vertex provides hints regarding symmetrical properties at $x = 5$ ). Solving gives $p = 5$ .

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 of the form $ax^2 + bx + c$ is given by the formula:

$x = -\frac{b}{2a}$
For $k(x)$ , we have $a = 1$ and $b = -10$ . Substituting gives:

$x = -\frac{-10}{2(1)} = 5$
Thus, the equation of the axis of symmetry is $x = 5$ .

The graphs of two functions, $f$ and $g$, are shown on the co-ordinate grid below - Junior Cycle Mathematics - Question 11 - 2015

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 - 2015

Match the graphs to the functions by writing $f$ or $g$ beside the corresponding graph on the grid.

Write down the roots of $f$ and the roots of $g$.

Sketch the graph of $h: x \mapsto (x - 1)^2 - 4$ on the co-ordinate grid above, where $x \in \mathbb{R}$.

Find the value of $p$.

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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