Math You Need > Graphing > Reading a point from a curve > Reading points - Practice problems

Reading a point from a curve or line
Practice Problems

The problems below walk you through the steps for reading points from a line. You can click on any of the graphs to open a bigger version or you can click the link under the graph to download a pdf of the graph for printing! Please try to complete these activities without peeking at the answers - this will help you when you get to the quiz at the end!

Radioactive Decay and Radiometric Dating

Geologists use information about the ratio of radioactive (parent) atoms to their decay product (daughter) atoms to understand the age of igneous, metamorphic and sedimentary rocks. The behavior of all radioactive elements is the same and the time it takes for one-half of the parent atoms to decay to daughter atoms is called the half-life. The graph below shows a plot of Daughter/Parent Ratio to Half-lives elapsed showing how geologists use isotopes to determine the age of rocks. This is a general plot that works for any isotope system. Use this plot to answer the questions below about reading points from a line.

Click to enlarge, or download a PDF (Acrobat (PDF) 25kB Jun25 10)

Question 1: Using the isotope plot above, determine the number of half-lives elapsed when the Daughter/Parent Ratio is 20.

  1. Begin by making sure you know what the axes represent. This is the most important step. Axes should be labeled with both numbers and the type of data plotted.
    Begin by looking at the axes. On the plot for this problem, the x-axis (horizontal axis) is labeled Half-lives Elapsed. The y-axis (vertical axis) is labeled Daughter/Parent Ratio.
  2. Look at the scale of each of the axes. In some cases, the scales may be the same; other times, they may have different scales (e.g., logarithmic, exponential, etc.).
    The axes on the plot for this problem are linear but each increment is different:

    On the x-axis of this plot, each (vertical) bold line represents 1 (half-life). Between each bold line, there are ten increments; these represent 0.1 half-lives each.

    On the y-axis, each bold (horizontal) line represents 20 (note that because Daughter/Parent is a ratio, there are no units on this axis). Between each bold line, there are ten increments; thus, the y-axis minor divisions are 2.
  3. Next, determine which axis shows the data you have been given in the question. Find the given value on that axis.
    In the question, you are told that the Daughter/Parent ratio is 20. Which of the axes provides information? The y-axis! Find 20 on the y-axis.

    See the red arrow pointing to the major division next to 20? You can click on the plot to make it bigger. This is a relatively easy number to find because 20 is one of the major divisions on the y-axis!

    finding y radiometric 1
  4. Follow the line representing your known data until it runs into the curve that you are interested in.
    In this case, you know the y value and you are asked to read from the isotope ratio curve (the blue line). Follow the horizontal line from 20 until it intersects the blue line.


    The red line extending from the red arrow shows where the curve intersects 20 on the y-axis (click to enlarge).

    reading over to curve
  5. Now, read the appropriate value from the other axis.

    Because you know the y-value, you want to read off the x-axis (which happens to be in units of half-life). Draw a line straight down until it intersects the x-axis (the green line on the plot shows you how to do this). Read the number that is there (the green arrow is pointing to the value - 4.4 half-lives. Remember that each increment on the x-axis is 0.1 half life. The line intersects the x-axis 4 units past 4 half-lives, so 4.4 half-lives must have elapsed.

    reading other axis 1

  6. You've now determined the answer to the problem!
    When Daughter/Parent Ratio is equal to 20, 4.4 half-lives have elapsed.Congratulations!

Question 2: Using the same isotope plot above, determine the Daughter/parent ratio when 5.5 half-lives have elapsed.

  1. Begin by making sure you know what the axes represent. If you completed the problem above, you should already know the answer to this step in the question, but just in case, you can peek below.
    Begin by looking at the axes. On the plot for this problem, the x-axis (horizontal axis) is labeled Half-lives Elapsed. The y-axis (vertical axis) is labeled Daughter/Parent Ratio.
  2. Look at the scale of each of the axes. Again, the scale is the same as in question 1 above, but here's the explanation again, just in case you don't remember.
    The axes on the plot for this problem are linear but each increment is different:

    On the x-axis of this plot, each (vertical) bold line represents 1 (half-life). Between each bold line, there are ten increments; these represent 0.1 half-lives each.

    On the y-axis, each bold (horizontal) line represents 20 (note that because Daughter/Parent is a ratio, there are no units on this axis). Between each bold line, there are ten increments; thus, the y-axis minor divisions are 2.
  3. Next, determine which axis shows the data you have been given in the question. Find the given value on that axis.
    In this question, you are asked to determine the Daughter/Parent ratio if you know the number of half-lives that have elapsed. Which of the axes provides the number of half lives? The x-axis! Find 5.5 on the x-axis.

    See the green arrow pointing to the x-axis? You can click on the plot to make it bigger. Remember that the x-axis is divided into increments of 0.1 half-lives; so, you should mark the spot 5 minor increments away from the number 5.

    reading known axis 2
  4. Follow the line representing your known data until it runs into the curve that you are interested in.
    In this case, you know the x value and you are asked to read the parent daughter ratio from the isotope ratio curve (the blue line). Follow a vertical line from 5.5 half-lives until it intersects the blue line.


    The green line extending from the green arrow shows where the curve intersects 5.5 on the x-axis (click to enlarge).

    reading up to curve 2
  5. Now, read the appropriate value from the other axis.

    Because you know the x-value, you want to read off the y-axis (which happens to be a ratio which has no units). Draw a line straight across until it intersects the y-axis (the red line on the plot shows you how to do this). Read the number that is there (the red arrow is pointing to the value - a ratio of 44. Remember that each increment on the y-axis is 2. The line intersects the y-axis 2 units above 40, so the ratio must be 44.

    reading across to y 2

  6. You've now determined the answer to the problem!
    When 5.5 half lives have elapsed, the Daughter/Parent Ratio is equal to 44.Congratulations!



Floods and Flood Frequency

Geologists keep track of the "stage" or height of floods every year and use that data to predict the probability that a flood will occur in any given year. The plot below is called a flood frequency curve, constructed from data collected over a number of years (sometimes as many as 100 -200 years. The plot below represents data from a hypothetical river for which we had 69 years of data. The probability is reported as something called a Recurrence interval and is reported in "years". Use the plot below to answer the following questions.
Click to enlarge or you can download a PDF version to print (Acrobat (PDF) 53kB Jun25 10).

Question 1: Determine the flood stage for a flood with a recurrence interval of 400 years.


  1. Begin by making sure you know what the axes represent. This is an extremely important step. Axes should be labeled with both numbers and the type of data plotted.
    Begin by looking at the axes. On the plot for this problem, the x-axis (horizontal axis) is labeled Recurrence Interval (years). The y-axis (vertical axis) is labeled Flood Stage (feet).
  2. Look at the scale of each of the axes. In some cases, the scales may be the same; other times, they may have different scales (e.g., logarithmic, exponential, etc.).
    The axes on the plot for this problem are different. The x-axis is a logartihmic scale (meaning that major divisions are in 10x increments. The y-axis on this plot is the more familiar linear axis. This kind of plot is called a semi-log plot because only ONE of the axes is logarithmic.

    On the x-axis of this plot, each (vertical) bold line represents 10 x (that is 100 = 1, 101 = 10, 102 = 100, etc.). Between each bold line, there are ten increments in varying sizes; these represent multiples of 1 (or 10 or 100). So, for example between 1 and 10, there are 10 increments so each gray line represents 2, 3, 4, 5, etc. Between 10 and 100, the minor divisions represent 20, 30, 40 and so on.

    On the y-axis, each bold (horizontal) line represents 5 feet (note that the bottom of the y-axis is not 0 - it starts at 10 feet). Between each bold line, there are five increments; thus, the y-axis minor divisions are 1 foot.
  3. Next, determine which axis shows the data you have been given in the question. Find the given value on that axis.
    In this question, you are given the recurrence interval (400 years). Which of the axes provides information about recurrence interval? The x-axis! Find 400 on the x-axis.

    See the orange arrow pointing to the x-axis? You can click on the plot to make it bigger. Remember that the x-axis is logarithmic and divisions are by 100's in this part of the plot. You should mark a point 3 increments past 100.

    Flood Frequency Curve read x
  4. Follow the line representing your known data until it runs into the curve that you are interested in.
    In this case, you know the x value and you are asked to read the flood stage from the flood frequency curve (the red line). Follow a vertical line from 400 years until it intersects the red line.


    The orange line extending from the orange arrow shows where the curve intersects 400 on the x-axis (click to enlarge).

    Flood Frequency Curve read up to curve 1
  5. Now, read the appropriate value from the other axis.

    Because you know the x-value, you want to read off the y-axis (recurrence interval in years). Draw a line straight across until it intersects the y-axis (the blue line on the plot shows you how to do this). Read the number that is there (the blue arrow is pointing to the value - a flood stage of 46 feet). The line intersects the y-axis 1 unit above 45 feet, so the flood stage must be 46 feet.

    Flood Frequency Curve read over to y 1

  6. You've now determined the answer to the problem!
    A flood stage of 46 feet represents a flood with a recurrence interval of 400 years on this river.Congratulations!


Question 2: Determine the recurrence interval for a flood stage of 38 feet.


  1. Begin by making sure you know what the axes represent. The axes haven't changed since the last problem but you can check below for information.
    On the plot for this problem, the x-axis (horizontal axis) is labeled Recurrence Interval (years). The y-axis (vertical axis) is labeled Flood Stage (feet).
  2. Look at the scale of each of the axes. The scale is the same as in Question 1 but you can make sure you still understand by checking below.
    The axes on the plot for this problem are different. The x-axis is a logartihmic scale (meaning that major divisions are in 10x increments. The y-axis on this plot is the more familiar linear axis. This kind of plot is called a semi-log plot because only ONE of the axes is logarithmic.

    On the x-axis of this plot, each (vertical) bold line represents 10 x (that is 100 = 1, 101 = 10, 102 = 100, etc.). Between each bold line, there are ten increments in varying sizes; these represent multiples of 1 (or 10 or 100). So, for example between 1 and 10, there are 10 increments so each gray line represents 2, 3, 4, 5, etc. Between 10 and 100, the minor divisions represent 20, 30, 40 and so on.

    On the y-axis, each bold (horizontal) line represents 5 feet (note that the bottom of the y-axis is not 0 - it starts at 10 feet). Between each bold line, there are five increments; thus, the y-axis minor divisions are 1 foot.
  3. Next, determine which axis shows the data you have been given in the question. Find the given value on that axis.
    In this question, you are given the flood stage (38 feet). Which of the axes provides information about flood stage? The y-axis! Find 38 on the y-axis.

    See the blue arrow pointing to the y-axis? You can click on the plot to make it bigger. Remember that the y-axis is divided into increments of 1 foot; so, you should mark the spot 3 minor increments away from the number 35.

    Flood Frequency Curve with Y marked
  4. Follow the line representing your known data until it runs into the curve that you are interested in.
    In this case, you know the y value and you are asked to read the recurrence interval from the flood frequency curve (the red line). Follow a horizontal line from 38 feet until it intersects the red line.


    The blue line extending from the blue arrow shows where the curve intersects 38 on the y-axis (click to enlarge).

    Flood Frequency Curve reading across
  5. Now, read the appropriate value from the other axis.

    Because you know the y-value, you want to read off the x-axis (recurrence interval in years). Draw a line straight down until it intersects the x-axis (the orange line on the plot shows you how to do this). Read the number that is there (the orange arrow is pointing to the value - a recurrence interval of 80 years. Check above in number 2 to check what the increments on the x-axis are (here they are in 10's). The line intersects the y-axis 3 units above 50 years, so the recurrence must be 80 years.

    Flood Frequency Curve reading down to x

  6. You've now determined the answer to the problem!
    When a flood stage of 38 feet is reached, the recurrence interval is 80 years.Congratulations!

Climate Change and Greenhouse Gases

Geoscientists use information gathered from the atmosphere and ice cores to understand long-term climate change and the role of greenhouse gases. On top of Mauna Loa, in Hawaii, a weather station collects information about the CO2 content of our atmosphere - Mauna Loa is far above the immediate influence of CO2 emissions from traffic because it is nearly 4170 meters (about 13680 feet) above sea level (see USGS Mauna Loa Volcano). The data collected between 1987 and 2006 is presented below in graphical form. Use this graph (you can download a larger version as a PDF to print) to answer the questions below. Note that the data shows a cyclical pattern that is associated with seasons (e.g., winter and summer) but that the trend shows that CO2 concentrations in our atmosphere are generally increasing.
Click to enlarge or you can download a PDF version to print (Acrobat (PDF) 43kB Jun28 10).


Question 1: What was the concentration of CO2 at Mauna Loa in January of the year 2000?


  1. Begin by making sure you know what the axes represent. This is an extremely important step. Axes should be labeled with both numbers and the type of data plotted.
    Begin by looking at the axes. On the plot for this problem, the x-axis (horizontal axis) is labeled Date (month and year). The y-axis (vertical axis) is labeled Carbon Dioxide (in ppm). (Note: ppm stands for parts per million)
  2. Look at the scale of each of the axes. In some cases, the scales may be the same; other times, they may have different scales (e.g., logarithmic, exponential, etc.).
    The axes on the plot for this problem are both linear axes.

    On the x-axis of this plot, each (vertical) bold line represents 6 months. Between each bold line, there are six increments representing minor divisions of one month. The entire data set spans 10 years and every January and July are marked.

    On the y-axis, each bold (horizontal) line represents 5 ppm (note that the bottom of the y-axis is not 0 - it starts at 345 ppm). Between each bold line, there are five increments; thus, the y-axis minor divisions are 1 ppm.
  3. Next, determine which axis shows the data you have been given in the question. Find the given value on that axis.
    In this question, you are given a date (January 2000). Which of the axes provides information about dates? The x-axis! Find Jan-00 on the x-axis.

    See the red arrow pointing to the x-axis? You can click on the plot to make it bigger. The red arrow is pointing to the major division of January 2000.

    Mauna Loa Jan 00 - find x
  4. Follow the line representing your known data until it runs into the curve that you are interested in.
    In this case, you know the x value and you are asked to read the carbon dioxide concentration from the curve (the orange line). Follow a vertical line from Jan-00 until it intersects the orange line.


    The red line extending from the red arrow shows where the curve intersects Jan-00 on the x-axis (click to enlarge).

    Mauna Loa Jan 00 - up to curve
  5. Now, read the appropriate value from the other axis.

    Because you know the x-value, you want to read off the y-axis (carbon dioxide in ppm). Draw a line straight across until it intersects the y-axis (the green line on the plot shows you how to do this). Read the number that is there (the green arrow is pointing to the value - a carbon dioxide concentration of 368.5 ppm). The line intersects the y-axis between 3 and 4 units above 365 ppm, so the concentration is approximately 368.5 ppm.

    Mauna Loa Jan 00 - over to y

  6. You've now determined the answer to the problem!
    In January 2000, sensors at the top of Mauna Loa detected about 368.5 ppm of carbon dioxide in the atmosphere.Congratulations!


Question 2: Determine what dates in the last 10 years (of record) showed that carbon dioxide concentrations were equal to 350 ppm.


  1. Begin by making sure you know what the axes represent. The axes haven't changed since the last problem but you can check below for information.
    On the plot for this problem, the x-axis (horizontal axis) is labeled Date (month and year). The y-axis (vertical axis) is labeled Carbon Dioxide (in ppm). (Note: ppm stands for parts per million)
  2. Look at the scale of each of the axes. The scale is the same as in Question 1 but you can make sure you still understand by checking below.
    The axes on the plot for this problem are both linear axes.

    On the x-axis of this plot, each (vertical) bold line represents 6 months. Between each bold line, there are six increments representing minor divisions of one month. The entire data set spans 10 years and every January and July are marked.

    On the y-axis, each bold (horizontal) line represents 5 ppm (note that the bottom of the y-axis is not 0 - it starts at 345 ppm). Between each bold line, there are five increments; thus, the y-axis minor divisions are 1 ppm.
  3. Next, determine which axis shows the data you have been given in the question. Find the given value on that axis.
    In this question, you are given a concentration and asked to find the date(s) when this concentration was recorded. Which of the axes provides information about carbon dioxide concentrations? The y-axis! Find 350 on the y-axis.

    See the green arrow pointing to the y-axis? You can click on the plot to make it bigger. Because the increments are in multiples of 5, 350 is marked and relatively easy to find.

    Mauna Loa 350 - marked y
  4. Follow the line representing your known data until it runs into the curve that you are interested in.
    In this case, you know the y value and you are asked to read the recurrence interval from the flood frequency curve (the red line). Follow a horizontal line from 38 feet until it intersects the orange curve.


    The green line extending from the green arrow shows where values of 350 occur on the curve (click to enlarge). Note that there are seven points at which the carbon dioxide concentrations were 350 ppm.

    Mauna Loa values of 350
  5. Now, read the appropriate values from the other axis.

    Because you know the y-value, you want to read off the x-axis (date in month and year). Draw lines from each point where the orange curve crosses 350 straight down until they intersect the x-axis (the red lines on the plot show you how to do this). Read the numbers keeping in mind that each minor increment is one month (the red arrows are pointing to appropriate values - Apr-87, Jul-87, Jan-88, Sep-88, Nov-88, Sep 89, Nov-89).

    Mauna Loa  x dates

  6. You've now determined the answer to the problem!
    Carbon dioxide concentrations were 350 ppm in the months of April and July 1987; January, September and November 1988; and September and November 1989.Congratulations!


Next Steps


Think you understand? Then go to wamap.org to take the online quiz.

If you still need help, you can go back to the explanation page or look at some of the links below.

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