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Grade 6 Math Circles
April 7, 2021
Graphing Functions — Solutions

Problem Set Solutions

To graph functions, it’s easiest to work by hand on paper (or on a tablet or touchscreen). You should draw and label the \(x\) and \(y\) axes, as well as mark down numbers along each axis (like on a number line). You can then plot points and graph functions on this Cartesian Plane. This will be the most straightforward with graph paper! At the end of this PDF is a sheet of graph paper which you can download to draw on or print out.

It is helpful to have a sense of what the \(x\) and \(y\) values will be before you begin graphing. Refer to ://www.cemc.uwaterloo.ca/events/mathcircles/2020-21/Winter/Junior6_Functions_Mar31_Solutions.pdfthis PDF, which contains tables for Questions 4–7 filled in for \(x=-3\) to \(x=3\). You can make the graph as big or as small as you need to fit a reasonable amount of the function! Here is an example, drawn by hand on graph paper.

\(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\)
\(y = f(x) = x + 3\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\)

With the help of this table, we draw the \(x\)- and \(y\)- axis of the Cartesian Plane to fit the points of our graph. Don’t forget to label the axes! We can then add points at the coordinates we’ve calculated, which in this case are \((-3, 0)\), \((-2, 1)\), \((-1, 2)\), \((0, 3)\), \((1, 4)\), \((2, 5)\), and \((3, 6)\). Lastly, we connect our dots to reveal the graph of this linear equation!

  1. Identify the coordinates of the six points plotted below:

    The coordinate grid. The horizontal axis starts at -5 and increases to 5. The vertical axis starts at -3 and increases to 3. There are 6 points, and a description of them follows.

    The same coordinate grid with the same points plotted. Each point is now labelled with a coordinate, as described in the following list.

  2. Use the graph to identify the value of \(y\) for each of the given values of \(x\).

    1. What is \(y\) at \(x = -3, \, -2, \,-1, \,0, \,1, \,2, \,3\) for this function?

      The coordinate grid. The horizontal axis starts at negative 4 and increases to positive 4. The vertical axis starts at negative 4 and increases to positive 4. A line is drawn. The line passes through coordinates that correspond to values on the horizontal and vertical axes, as given in the following list.

      Solution:

      • at \(x=-3\), \(y=2.5\)

      • at \(x=-2\), \(y=2\)

      • at \(x=-1\), \(y=1.5\)

      • at \(x=0\), \(y=1\)

      • at \(x=1\), \(y=0.5\)

      • at \(x=2\), \(y=0\)

      • at \(x=3\), \(y=-0.5\)

    2. What is \(y\) at \(x = -4,\, -3,\, -2, \,-1,\, 0,\, 1,\, 2,\, 3, \,4\) for this function?

      The coordinate grid. The horizontal axis starts at negative 10 and increases to positive 10. The vertical axis starts at 0 and increases to positive 15. A parabola is drawn on the grid. The parabola passes through coordinates that correspond to values on the horizontal and vertical axes, as given in the following list.

      Solution:

      • at \(x=-4\), \(y=16\)

      • at \(x=-3\), \(y=9\)

      • at \(x=-2\), \(y=4\)

      • at \(x=-1\), \(y=1\)

      • at \(x=0\), \(y=0\)

      • at \(x=1\), \(y=1\)

      • at \(x=2\), \(y=4\)

      • at \(x=3\), \(y=9\)

      • at \(x=4\), \(y=16\)

    3. What is \(y\) at \(x = -1,\, 0, \,1, \,2\) for this function?

      The coordinate grid. The horizontal axis starts at negative 3 and increases to positive 3. The vertical axis starts at negative 3 and increases to positive 3. A curve is drawn on the grid. The curve passes through coordinates that correspond to values on the horizontal and vertical axes, as given in the following list.

      Solution:

      • at \(x=-1\), \(y=-1\)

      • at \(x=0\), \(y=0\)

      • at \(x=1\), \(y=-1\)

      • at \(x=2\), \(y=2\)

  3. At what values of \(x\) is the value of \(y\) equal to 1 for this function?

    The coordinate grid. The horizontal axis starts at negative 4 and increases to positive 4. The vertical axis starts at negative 4 and increases to positive 4. A curve is drawn on the grid. The curve passes through coordinates that correspond to values on the horizontal and vertical axes, as given in the following list.

    Solution: On the section of the function shown, \(y=1\) at \(x=-3.5, \, -1.5, \, 0.5, \, 2.5,\) and \(4.5\).

    As a general formula, \(y=1\) at \(x \in \{0.5 + 2n, \, n \in \mathbb{Z}\}\). That is, \(y=1\) at all values of \(x\) in the set of numbers that can be expressed as \(0.5+2n\), when \(n\) is an integer.

  4. Graph each function. Identify the domain and range.

    1. \(f(x) = x - 2\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\mathbb{R}\)

      The coordinate grid. The x-axis goes from negative 3 to 3. The y-axis goes from negative 3 to 3. The function of a straight line that goes through the coordinates (0, negative 2) and (2,0).

    2. \(g(x) = 3x - 5\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\mathbb{R}\)

      The coordinate grid. The x-axis goes from negative 6 to 6. The y-axis goes from negative 6 to 6. The function is a straight line that goes through the coordinates (0, negative 5) and (3,4).

    3. \(h(x) = -x + 25\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\mathbb{R}\)

      The coordinate grid. The x-axis goes from negative 40 to 40, increasing by 5s. The y-axis increases by 5s from negative 40 to 40. The function is a straight line that goes through the coordinates (0,25) and (25,0).

    4. \(j(x) = \frac{x + 5}{2}\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\mathbb{R}\)

      The coordinate grid. The x-axis increases by 0.5 and goes from negative 5.5 to 0.5. The y-axis increases by 0.5 and goes from negative 3 to 3.5. The function is a straight line that goes through the coordinates (negative 5,0) and (0, 2.5).

    5. \(k(x) = \frac{1}{4}x\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\mathbb{R}\)

      The coordinate grid. The x-axis goes from negative 5 to 5. The y-axis goes from negative 5 to 5. The function is a straight line that goes through the coordiantes (negative 4, negative 1) and (4,1).

    6. \(l(x) = 100 - 10x\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\mathbb{R}\)

      The coordinate grid. The x-axis goes from negative 100 to 100, increasing by 50. The y-axis also increases by 50 from negative 100 to 100. The function is a straight line that goes through the coordinates (0,100) and (20,negative 100).

  5. Graph each function. Identify the domain and range.

    1. \(f(x) = x^2 + 2\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\{y \in \mathbb{R}, \, y \geq 2\}\)

      The coordinate grid. The x-axis increases from negative 10 to 10. The y-axis increases from 0 to 20. The function is a parabola. The vertex has coordinates (0,2) and the parabola opens up. The parabola also passes through the coordinates: (negative 4, 18), (negative 3, 11), (negative 2, 6), (negative 1, 3), (1,3), (2, 6), (3,11) and (4,18).

    2. \(g(x) = \frac{x^2}{2}\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\{y \in \mathbb{R}, \, y \geq 0\}\)

      The coordinate grid. The x-axis increases from negative 5 to 5. The y-axis increases from negative 1 to 8. The function is a parabola. The vertex has coordinates (0,0) and the parabola opens up. The parabola also passes through the coordinates: (negative 4, 8), (negative 2, 2), (2,2) and (4,8).

    3. \(h(x) = -x^2\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\{y \in \mathbb{R}, \, y \leq 0\}\)

      The coordinate grid. The x-axis increases from negative 10 to 10. The y-axis goes from 0 to -15. The function is a parabola. The vertex has coordinates (0,0) and the parabola opens down. The parabola also passes through the coordinates: (negative 4, negative 16), (negative 3, negative 9), (negative 2, negative 4), (negative 1, negative 1), (1,negative 1), (2, negative 4), (3,negative 9) and (4,negative 16).

    4. \(j(x) = 2x^2\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\{y \in \mathbb{R}, \, y \geq 0\}\)

      The coordinate grid. The x-axis increases from negative 10 to 10. The y-axis increases from 0 to 20. The function is a parabola. The vertex has coordinates (0,0) and the parabola opens up. The parabola also passes through the coordinates: (negative 3, 18), (negative 2, 8), (negative 1, 2), (1,2), (2, 8), and (3,18).

    5. \(k(x) = 0.5x^2 + x\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\{y \in \mathbb{R}, \, y \geq -0.5\}\)

      The coordinate grid. The x-axis increases from negative 7 to 8. The y-axis increases from negative 2 to 13. The function is a parabola. The vertex has x-coordinate x equal negative 1 and the parabola opens up. The parabola also passes through the coordinates: (negative 6, 12), (negative 4, 4), (negative 2, 0), (0,0), (2, 4), and (4,12).

    6. \(l(x) = x^2 + 4x + 3\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\{y \in \mathbb{R}, \, y \geq -1\}\)

      The coordinate grid. The x-axis increases from negative 6 to 6. The y-axis increases from negative 2 to 10. The function is a parabola. The vertex has coordinates (negative 2, negative 1) and the parabola opens up. The parabola also passes through the coordinates: (negative 5, 8), (negative 4, 3), (negative 3, 0), (negative 1, 0), (0,3), and (1, 8).

  6. Graph each function. Identify the domain and range.

    1. \(f(x) = x^3\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\mathbb{R}\)

      The graph f(x)

    2. \(g(x) = x^4\)

      Solution: Domain: \(\mathbb{R}\); Range: \(\{y \in \mathbb{R}, \, y \geq 0\}\)

      The graph g(x)

  7. Graph each function. Identify any asymptotes. Identify the domain and range.

    1. \(f(x) = \frac{1}{x + 1}\)

      Solution: There is a vertical asymptote at \(x=-1\) and a horizontal asymptote at \(y=0\). Domain: \(\{x \in \mathbb{R}, \, x \neq -1\}\); Range: \(\{y \in \mathbb{R}, \, y \neq 0\}\)

      The graph f(x)

    2. \(g(x) = \frac{3}{2x}\)

      Solution: There is a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\). Domain: \(\{x \in \mathbb{R}, \, x \neq 0\}\); Range: \(\{y \in \mathbb{R}, \, y \neq 0\}\)

      The graph g(x)

    3. \(h(x) = \frac{x}{x - 4}\)

      Solution: There is a vertical asymptote at \(x=4\) and a horizontal asymptote at \(y=1\). Domain: \(\{x \in \mathbb{R}, \, x \neq 4\}\); Range: \(\{y \in \mathbb{R}, \, y \neq 1\}\)

      The graph h(x)

    4. \(j(x) = \frac{x+1}{x+2}\)

      Solution: There is a vertical asymptote at \(x=-2\) and a horizontal asymptote at \(y=1\). Domain: \(\{x \in \mathbb{R}, \, x \neq -2\}\); Range: \(\{y \in \mathbb{R}, \, y \neq 1\}\)

      The graph j(x)

    5. \(k(x) = \frac{5x}{5x + 2}\)

      Solution: There is a vertical asymptote at \(x=-\frac{2}{5}=-0.4\) and a horizontal asymptote at \(y=1\). Domain: \(\{x \in \mathbb{R}, \, x \neq -\frac{2}{5}\}\); Range: \(\{y \in \mathbb{R}, \, y \neq 1\}\).

      The graph k(x)

    6. \(l(x) = \frac{x^2}{x}\)

      Solution: There are no asymptotes, but there is a gap in the graph at the point \((0,0)\), because \(l(0)\) is undefined. Domain: \(\{x \in \mathbb{R}, \, x \neq 0\}\); Range: \(\{y \in \mathbb{R}, \, y \neq 0\}\).

      The graph l(x)

  8. Using the vertical line test, identify whether each of the following graphs represents a function.

    1. The coordinate grid with both x and y axes starting at negative 10 and increasing to 10. The graph is made up of two line segments. The first segment starts at (negative 10, 10) and ends at (0,0). The second segment starts at (0,0) and ends at (10,10).

    2. The coordinate grid with both x and y axes starting at negative 10 and increasing to 10. A curve starts at (0,0) and goes through (4,2) and (9,3).

    3. The coordinate grid with both x and y axes starting at negative 10 and increasing to 10. A circle that goes through the coordinates (negative 5,0) and (5,0) as well as (0, negative 5) and (0,5).

    4. The coordinate grid. The graph is split into 3 non-touching sub-graphs, two of which resemble the reciprocal function with vertical asymptotes at x=-1 and x=1. The third piece lies in between the two vertical asymptotes and passes through each vertical marking once.

    5. The coordinate grid. For each marking on the vertical axis, the negative square of that value is plotted onto the horizontal axis.

    6. The coordinate grid with both x and y axes starting at negative 4 and increasing to 4. Points are plotted at (negative 4, 4), (negative 3, 3), (negative 2, 2) and so on up (3,3), (4,4).

    1. yes

    2. yes

    3. yes

    4. no

    5. no

    6. yes

  9. Cam is opening a side business at home baking cakes. It takes them 1.5 hours to make each cake, plus 1 hour to clean up the kitchen at the end of each day if they choose to make any cakes that day. To make sure there’s still time for school, Cam can spend a maximum of 6 hours each day working on their business.

    1. Express how long it would take Cam to make \(n\) cakes in a day as a function. Use \(T\)as the function name (to represent “time” in hours), and use \(n\) as the variable name (to represent the “number of cakes”).

      Solution: \[T(n) = \begin{cases} 1.5n + 1 & \text{if $n > 0$} \\ 0 & \text{if $n = 0$} \\ \end{cases}\]

    2. Graph the function \(T(n)\).

      Solution:

      The coordinate grid. The x-axis increases by 0.5 from negative 1.5 to 5.5. The y-axis increases by 0.5 from negative 0.5 to 6.5. Four points are plotted at (0,0), (1,2.5), (2,4) and (3,5.5).

    3. Using the graph, determine the maximum number of cakes that Cam can make in a day.

      Solution: Cam can spend a maximum of 6 hours each day working on their business, so \(y\) must be 6 or less. Also, Cam can only make whole cakes—“half” a cake would make no sense! This means that \(x\) must be a non-negative integer. Thus, the maximum number of cakes that Cam can make in a day is 3 cakes, which would take them 5.5 hours.

    4. Taking into account the context of this function, what are the domain and range of \(T(n)\)?

    Solution: \(T(n)\) tells us how long it would take for Cam to make \(n\) cakes in a day. Thus, we take into account the limitation that \(n\) must be a whole number. However, even though Cam only has time to make 3 cakes in a day, it is still true that it would take \(1.5n+1\) hours to make any other positive integer number of cakes! Therefore, we have:

    Domain: \(\{n \in \mathbb{Z}, \, n \geq 0\}\); Range: \(\{T(n) = 0, \, 1.5n+1\}\) (which reads as \(T(n)\) equal to \(0\) or \(1.5n+1\)).

    Hint: A function’s graph doesn’t always have to be a fully filled-in line! As long as it passes the vertical line test, it’s a valid function.