{"id":1863,"date":"2016-06-28T04:25:35","date_gmt":"2016-06-28T04:25:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&p=1863"},"modified":"2017-08-16T03:35:10","modified_gmt":"2017-08-16T03:35:10","slug":"outcome-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/outcome-graphs-of-functions\/","title":{"raw":"Graphs of Functions","rendered":"Graphs of Functions"},"content":{"raw":"
\r\n

Learning Objectives<\/span><\/h2>\r\n
    \r\n \t
  • Graph linear functions using a table of values<\/li>\r\n \t
  • Graph a quadratic function using a table of values<\/li>\r\n \t
  • Identify important features of\u00a0the graphs of a quadratic functions of the form [latex]f(x)=ax^2+bx+c[\/latex]<\/li>\r\n \t
  • Graph a radical function using a table of values<\/li>\r\n \t
  • Identify\u00a0how multiplication can change the graph of a radical function<\/li>\r\n \t
  • Identify how addition and subtraction can change the graph of a radical function<\/li>\r\n \t
  • Define one-to-one function<\/li>\r\n \t
  • Use the horizontal\u00a0line test to determine whether a function is one-to-one<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen both the input (independent variable) and the output (dependent variable) are real numbers, a function can be represented by a coordinate graph. The input is plotted on the horizontal\u00a0x<\/i>-axis and the output is plotted on the vertical\u00a0y<\/i>-axis.\r\n\r\nA helpful first step in graphing a function is to make a table of values. This is particularly useful when you don\u2019t know the general shape the function will have. You probably already know that a linear function will be a straight line, but let\u2019s make a table first to see how it can be helpful.\r\n\r\nWhen making a table, it\u2019s a good idea to include negative values, positive values, and zero to ensure that you do have a linear function.\r\n\r\nMake a table of values for [latex]f(x)=3x+2[\/latex].\r\n\r\nMake a two-column table. Label the columns x<\/i> and f<\/i>(x<\/i>).\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/i><\/th>\r\nf<\/i>(x<\/i>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    <\/td>\r\n<\/td>\r\n<\/tr>\r\n
    <\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nChoose several values for x<\/i> and put them as separate rows in the x<\/i> column. These are YOUR CHOICE - there is no \"right\" or \"wrong\" values to pick, just go for it.\r\n\r\nTip:<\/i> It\u2019s always good to include 0, positive values, and negative values, if you can.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/i><\/th>\r\nf<\/i>(x<\/i>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]\u22122[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex]\u22121[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex]3[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEvaluate the function for each value of x<\/i>, and write the result in the f<\/i>(x<\/i>) column next to the x<\/i> value you used.\r\n\r\nWhen [latex]x=0[\/latex], [latex]f(0)=3(0)+2=2[\/latex],\r\n\r\n[latex]f(1)=3(1)+2=5[\/latex],\r\n\r\n[latex]f(\u22121)=3(\u22121)+2=\u22123+2=\u22121[\/latex],\u00a0and so on.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/i><\/td>\r\nf<\/i>(x<\/i>)<\/td>\r\n<\/tr>\r\n
    [latex]\u22122[\/latex]<\/td>\r\n[latex]\u22124[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\u22121[\/latex]<\/td>\r\n[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]3[\/latex]<\/td>\r\n[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n(Note that your table of values may be different from someone else\u2019s. You may each choose different numbers for x<\/i>.)\r\n\r\nNow that you have a table of values, you can use them to help you draw both the shape and location of the function. Important:<\/i> The graph of the function will show all possible values of x<\/i> and the corresponding values of y<\/i>. This is why the graph is a line and not just the dots that make up the points in our table.\r\n\r\nGraph [latex]f(x)=3x+2[\/latex].\r\nUsing the table of values we created above you can think of f<\/i>(x<\/i>) as y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/i><\/td>\r\nf<\/i>(x<\/i>)<\/td>\r\n<\/tr>\r\n
    [latex]\u22122[\/latex]<\/td>\r\n[latex]\u22124[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\u22121[\/latex]<\/td>\r\n[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]3[\/latex]<\/td>\r\n[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points.\r\n\"The\r\n\r\nSince the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.\r\n\r\n\"A\r\n\r\nLet\u2019s try another one. Before you look at the answer, try to make the table yourself and draw the graph on a piece of paper.\r\n
    \r\n

    Example<\/h3>\r\nGraph [latex]f(x)=\u2212x+1[\/latex].\r\n\r\n[reveal-answer q=\"748367\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"748367\"]Start with a table of values. You can choose different values for x<\/i>, but once again, it\u2019s helpful to include 0, some positive values, and some negative values.\r\n\r\nIf you think of f<\/i>(x<\/i>) as y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.\r\n

    [latex]f(\u22122)=\u2212(\u22122)+1=2+1=3\\\\f(\u22121)=\u2212(\u22121)+1=1+1=2\\\\f(0)=\u2212(0)+1=0+1=1\\\\f(1)=\u2212(1)+1=\u22121+1=0\\\\f(2)=\u2212(2)+1=\u22122+1=\u22121[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/i><\/td>\r\nf<\/i>(x<\/i>)<\/td>\r\n<\/tr>\r\n
    [latex]\u22122[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\u22121[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]2[\/latex]<\/td>\r\n[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points.\r\n\"The\r\n

    Answer<\/h4>\r\n\"Line\r\n\r\nSince the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to graph a linear function on a set of coordinate axes.\r\n\r\nhttps:\/\/youtu.be\/sfzpdThXpA8\r\n\r\nThese graphs are representations of a linear function. Remember that a function is a correspondence between two variables, such as x<\/i> and y<\/i>.\r\n
    \r\n

    A General Note: Linear Function<\/h3>\r\nA linear function<\/strong> is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line\r\n\r\n[latex]f\\left(x\\right)=mx+b[\/latex]\r\n\r\nwhere [latex]b[\/latex]\u00a0is the initial or starting value of the function (when input, [latex]x=0[\/latex]), and [latex]m[\/latex]\u00a0is the constant rate of change, or slope<\/strong> of the function. The y<\/em>-intercept<\/strong> is at [latex]\\left(0,b\\right)[\/latex]\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"
    \n

    Learning Objectives<\/span><\/h2>\n
      \n
    • Graph linear functions using a table of values<\/li>\n
    • Graph a quadratic function using a table of values<\/li>\n
    • Identify important features of\u00a0the graphs of a quadratic functions of the form [latex]f(x)=ax^2+bx+c[\/latex]<\/li>\n
    • Graph a radical function using a table of values<\/li>\n
    • Identify\u00a0how multiplication can change the graph of a radical function<\/li>\n
    • Identify how addition and subtraction can change the graph of a radical function<\/li>\n
    • Define one-to-one function<\/li>\n
    • Use the horizontal\u00a0line test to determine whether a function is one-to-one<\/li>\n<\/ul>\n<\/div>\n

      When both the input (independent variable) and the output (dependent variable) are real numbers, a function can be represented by a coordinate graph. The input is plotted on the horizontal\u00a0x<\/i>-axis and the output is plotted on the vertical\u00a0y<\/i>-axis.<\/p>\n

      A helpful first step in graphing a function is to make a table of values. This is particularly useful when you don\u2019t know the general shape the function will have. You probably already know that a linear function will be a straight line, but let\u2019s make a table first to see how it can be helpful.<\/p>\n

      When making a table, it\u2019s a good idea to include negative values, positive values, and zero to ensure that you do have a linear function.<\/p>\n

      Make a table of values for [latex]f(x)=3x+2[\/latex].<\/p>\n

      Make a two-column table. Label the columns x<\/i> and f<\/i>(x<\/i>).<\/p>\n\n\n\n\n\n\n
      x<\/i><\/th>\nf<\/i>(x<\/i>)<\/th>\n<\/tr>\n<\/thead>\n
      <\/td>\n<\/td>\n<\/tr>\n
      <\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Choose several values for x<\/i> and put them as separate rows in the x<\/i> column. These are YOUR CHOICE – there is no “right” or “wrong” values to pick, just go for it.<\/p>\n

      Tip:<\/i> It\u2019s always good to include 0, positive values, and negative values, if you can.<\/p>\n\n\n\n\n\n\n\n\n\n
      x<\/i><\/th>\nf<\/i>(x<\/i>)<\/th>\n<\/tr>\n<\/thead>\n
      [latex]\u22122[\/latex]<\/td>\n<\/td>\n<\/tr>\n
      [latex]\u22121[\/latex]<\/td>\n<\/td>\n<\/tr>\n
      [latex]0[\/latex]<\/td>\n<\/td>\n<\/tr>\n
      [latex]1[\/latex]<\/td>\n<\/td>\n<\/tr>\n
      [latex]3[\/latex]<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Evaluate the function for each value of x<\/i>, and write the result in the f<\/i>(x<\/i>) column next to the x<\/i> value you used.<\/p>\n

      When [latex]x=0[\/latex], [latex]f(0)=3(0)+2=2[\/latex],<\/p>\n

      [latex]f(1)=3(1)+2=5[\/latex],<\/p>\n

      [latex]f(\u22121)=3(\u22121)+2=\u22123+2=\u22121[\/latex],\u00a0and so on.<\/p>\n\n\n\n\n\n\n\n\n
      x<\/i><\/td>\nf<\/i>(x<\/i>)<\/td>\n<\/tr>\n
      [latex]\u22122[\/latex]<\/td>\n[latex]\u22124[\/latex]<\/td>\n<\/tr>\n
      [latex]\u22121[\/latex]<\/td>\n[latex]\u22121[\/latex]<\/td>\n<\/tr>\n
      [latex]0[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
      [latex]1[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n<\/tr>\n
      [latex]3[\/latex]<\/td>\n[latex]11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      (Note that your table of values may be different from someone else\u2019s. You may each choose different numbers for x<\/i>.)<\/p>\n

      Now that you have a table of values, you can use them to help you draw both the shape and location of the function. Important:<\/i> The graph of the function will show all possible values of x<\/i> and the corresponding values of y<\/i>. This is why the graph is a line and not just the dots that make up the points in our table.<\/p>\n

      Graph [latex]f(x)=3x+2[\/latex].
      \nUsing the table of values we created above you can think of f<\/i>(x<\/i>) as y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.<\/p>\n\n\n\n\n\n\n\n\n
      x<\/i><\/td>\nf<\/i>(x<\/i>)<\/td>\n<\/tr>\n
      [latex]\u22122[\/latex]<\/td>\n[latex]\u22124[\/latex]<\/td>\n<\/tr>\n
      [latex]\u22121[\/latex]<\/td>\n[latex]\u22121[\/latex]<\/td>\n<\/tr>\n
      [latex]0[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
      [latex]1[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n<\/tr>\n
      [latex]3[\/latex]<\/td>\n[latex]11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Plot the points.
      \n\"The<\/p>\n

      Since the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.<\/p>\n

      \"A<\/p>\n

      Let\u2019s try another one. Before you look at the answer, try to make the table yourself and draw the graph on a piece of paper.<\/p>\n

      \n

      Example<\/h3>\n

      Graph [latex]f(x)=\u2212x+1[\/latex].<\/p>\n

      Show Solution<\/span><\/p>\n
      Start with a table of values. You can choose different values for x<\/i>, but once again, it\u2019s helpful to include 0, some positive values, and some negative values.<\/p>\n

      If you think of f<\/i>(x<\/i>) as y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.<\/p>\n

      [latex]f(\u22122)=\u2212(\u22122)+1=2+1=3\\\\f(\u22121)=\u2212(\u22121)+1=1+1=2\\\\f(0)=\u2212(0)+1=0+1=1\\\\f(1)=\u2212(1)+1=\u22121+1=0\\\\f(2)=\u2212(2)+1=\u22122+1=\u22121[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
      x<\/i><\/td>\nf<\/i>(x<\/i>)<\/td>\n<\/tr>\n
      [latex]\u22122[\/latex]<\/td>\n[latex]3[\/latex]<\/td>\n<\/tr>\n
      [latex]\u22121[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
      [latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n<\/tr>\n
      [latex]1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n<\/tr>\n
      [latex]2[\/latex]<\/td>\n[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Plot the points.
      \n\"The<\/p>\n

      Answer<\/h4>\n

      \"Line<\/p>\n

      Since the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.\n<\/p><\/div>\n<\/div>\n<\/div>\n

      In the following video we show another example of how to graph a linear function on a set of coordinate axes.<\/p>\n