{"id":819,"date":"2025-07-15T19:38:31","date_gmt":"2025-07-15T19:38:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=819"},"modified":"2026-03-05T19:17:16","modified_gmt":"2026-03-05T19:17:16","slug":"linear-models-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-models-learn-it-5\/","title":{"raw":"Linear Models: Learn It 5","rendered":"Linear Models: Learn It 5"},"content":{"raw":"

Understanding Interpolation and Extrapolation<\/h2>\r\nWhile the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as interpolation<\/strong> when we predict a value inside the domain and range of the data. The process of extrapolation<\/strong> is used when we predict a value outside the domain and range of the data.\r\n\r\n
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interpolation and extrapolation<\/h3>\r\nDifferent methods of making predictions are used to analyze data.\r\n
    \r\n \t
  • The method of interpolation<\/strong> involves predicting a value inside the domain and\/or range of the data.<\/li>\r\n \t
  • The method of extrapolation<\/strong> involves predicting a value outside the domain and\/or range of the data.<\/li>\r\n \t
  • Model breakdown<\/strong> occurs at the point when the model no longer applies.<\/li>\r\n<\/ul>\r\n<\/section>
    \"ScatterThe graph compares the two processes for the cricket-chirp data addressed in the previous\u00a0example.We can see that interpolation<\/strong> would occur if we used our model to predict temperature when the values for chirps are between [latex]18.5[\/latex] and [latex]44[\/latex].Extrapolation<\/strong> would occur if we used our model to predict temperature when the values for chirps are less than [latex]18.5[\/latex] or greater than [latex]44[\/latex].Previously, we have found that the estimated equation for the line is [latex]y = 1.12x+31.22[\/latex].Use the cricket data above\u00a0to answer the following questions:\r\n
      \r\n \t
    1. Would predicting the temperature when crickets are chirping [latex]30[\/latex] times in [latex]15[\/latex] seconds be interpolation or extrapolation? Make the prediction and discuss whether it is reasonable.<\/li>\r\n \t
    2. Would predicting the number of chirps crickets will make at [latex]40[\/latex] degrees be interpolation or extrapolation? Make the prediction and discuss whether it is reasonable.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"281433\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"281433\"]\r\n
        \r\n \t
      1. Since [latex]30[\/latex] chirps is between the values of [latex]18.5[\/latex] and [latex]44[\/latex] chirps, predicting the temperature for [latex]30[\/latex] chirps is an interpolation.\r\n
        [latex]\\begin{align*} y &= 1.12(30) + 31.22 \\\\ y &= 33.6 + 31.22 \\\\ y &= 64.82 \\end{align*}[\/latex]<\/center>So, the predicted temperature is [latex]64.82[\/latex] degrees Farenheit.<\/strong>\r\n[latex]\\\\[\/latex]\r\nThis prediction is reasonable because it falls within the range of the data we used to create the model. Interpolation generally provides more reliable predictions as it works within the range of observed values.<\/li>\r\n \t
      2. The temperature values varied from [latex]52[\/latex] to [latex]80.5[\/latex]. Predicting the number of chirps at [latex]40[\/latex] degrees is extrapolation because [latex]40[\/latex] is outside the range of our data.\r\n
        [latex]\\begin{align*} 40 &= 1.12x + 31.22 \\\\ 40 - 31.22 &= 1.12x \\\\ 8.78 &= 1.12x \\\\ x &= \\frac{8.78}{1.12} \\\\ x &\\approx 7.84 \\end{align*}[\/latex]<\/center>So, the predicted number of chirps is approximately [latex]7.84[\/latex]<\/strong>.This prediction may be less reliable because it involves extrapolation. Extrapolation can lead to less accurate predictions since it extends beyond the range of observed data.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section>
        Interpolation<\/strong> occurs within the domain and range of the provided data whereas extrapolation<\/strong> occurs outside.<\/section>
        [ohm_question hide_question_numbers=1]318768[\/ohm_question]<\/section>","rendered":"

        Understanding Interpolation and Extrapolation<\/h2>\n

        While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as interpolation<\/strong> when we predict a value inside the domain and range of the data. The process of extrapolation<\/strong> is used when we predict a value outside the domain and range of the data.<\/p>\n

        \n

        interpolation and extrapolation<\/h3>\n

        Different methods of making predictions are used to analyze data.<\/p>\n

          \n
        • The method of interpolation<\/strong> involves predicting a value inside the domain and\/or range of the data.<\/li>\n
        • The method of extrapolation<\/strong> involves predicting a value outside the domain and\/or range of the data.<\/li>\n
        • Model breakdown<\/strong> occurs at the point when the model no longer applies.<\/li>\n<\/ul>\n<\/section>\n
          \"ScatterThe graph compares the two processes for the cricket-chirp data addressed in the previous\u00a0example.We can see that interpolation<\/strong> would occur if we used our model to predict temperature when the values for chirps are between [latex]18.5[\/latex] and [latex]44[\/latex].Extrapolation<\/strong> would occur if we used our model to predict temperature when the values for chirps are less than [latex]18.5[\/latex] or greater than [latex]44[\/latex].Previously, we have found that the estimated equation for the line is [latex]y = 1.12x+31.22[\/latex].Use the cricket data above\u00a0to answer the following questions:<\/p>\n
            \n
          1. Would predicting the temperature when crickets are chirping [latex]30[\/latex] times in [latex]15[\/latex] seconds be interpolation or extrapolation? Make the prediction and discuss whether it is reasonable.<\/li>\n
          2. Would predicting the number of chirps crickets will make at [latex]40[\/latex] degrees be interpolation or extrapolation? Make the prediction and discuss whether it is reasonable.<\/li>\n<\/ol>\n