Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$ in the domain $-2 ext{ ≤ } x ext{ ≤ } 4$, where $x ext{ ∈ } ext{ℝ}$. There is more room for working out on the next page. The function $f : x ightarrow x^2 - 2x + 4$ gives the predicted wind speed, in km per hour, over a 6-hour period of time. The x-axis represents the time from 10 p.m. ($x = -2$) to 4 a.m. ($x = 4$). Use your graph from (i) to answer the following questions. Show your work on the graph. What is the predicted wind speed at midnight? Find the times when the predicted wind speed is 5-5 km per hour. If the wind speed is between 1-1 km per hour and 5-5 km per hour, it is called light air. According to your graph, for how long will the wind be light air?

Junior Cycle Mathematics Functions & Graphs: Draw the graph of the function $

Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$ in the domain $-2 ext{ ≤ } x ext{ ≤ } 4$, where $x ext{ ∈ } ext{ℝ}$ - Junior Cycle Mathematics - Question 15 - 2014

Question 15

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Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$ in the domain $-2 ext{ ≤ } x ext{ ≤ } 4$, where $x ext{ ∈ } ext{ℝ}$. There is more room for workin... show full transcript

Worked Solution & Example Answer:Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$ in the domain $-2 ext{ ≤ } x ext{ ≤ } 4$, where $x ext{ ∈ } ext{ℝ}$ - Junior Cycle Mathematics - Question 15 - 2014

Step 1

Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$

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Answer

To draw the graph of the quadratic function, we need to find the vertex and some key points in the given domain. The vertex can be found using the vertex formula

$x = -\frac{b}{2a} = -\frac{-2}{2(1)} = 1$

Substituting back into the function:

$f(1) = (1)^2 - 2(1) + 4 = 3.$

Thus, the vertex is at (1, 3). We calculate a few more points:

$f(-2) = (-2)^2 - 2(-2) + 4 = 12$
$f(-1) = (-1)^2 - 2(-1) + 4 = 7$
$f(0) = (0)^2 - 2(0) + 4 = 4$
$f(2) = (2)^2 - 2(2) + 4 = 4$
$f(3) = (3)^2 - 2(3) + 4 = 7$
$f(4) = (4)^2 - 2(4) + 4 = 12$

Plotting these points gives us the graph with the vertex located at (1, 3) and symmetric points calculated as shown.

Step 2

What is the predicted wind speed at midnight?

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Answer

Midnight corresponds to $x = 0$ . From the function, we calculate:

$f(0) = 4.$

Thus, the predicted wind speed at midnight is 4 km/h.

Step 3

Find the times when the predicted wind speed is 5-5 km per hour.

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Answer

To find when the wind speed is 5 km/h, we set the function equal to 5:

$x^2 - 2x + 4 = 5$

Solving,

$x^2 - 2x - 1 = 0$

Using the quadratic formula,

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)} = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}.$

The approximate values are:

$x \approx 2.4$ (at 11:25 p.m.)
$x \approx -0.4$ (at 2:35 a.m.)

Step 4

If the wind speed is between 1-1 km per hour and 5-5 km per hour, it is called light air. According to your graph, for how long will the wind be light air?

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Answer

The range for light air is defined as between 1 km/h and 5 km/h. From the graph, we can see that this occurs between the times when the function intersects with the lines $y=1$ and $y=5$ . Assuming these correspond to $x ext{ intervals of } [0, 2]$ and checking values yields that the light air persists approximately from 2 hours to 3 hours and 10 minutes, making it about 3 hours and 10 minutes.

Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$ in the domain $-2 ext{ ≤ } x ext{ ≤ } 4$, where $x ext{ ∈ } ext{ℝ}$ - Junior Cycle Mathematics - Question 15 - 2014

Worked Solution & Example Answer:Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$ in the domain $-2 ext{ ≤ } x ext{ ≤ } 4$, where $x ext{ ∈ } ext{ℝ}$ - Junior Cycle Mathematics - Question 15 - 2014

Draw the graph of the function $f : x ightarrow x^2 - 2x + 4$

What is the predicted wind speed at midnight?

Find the times when the predicted wind speed is 5-5 km per hour.

If the wind speed is between 1-1 km per hour and 5-5 km per hour, it is called light air. According to your graph, for how long will the wind be light air?

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