{"id":154,"date":"2023-09-20T22:47:55","date_gmt":"2023-09-20T22:47:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/transformations-of-functions\/"},"modified":"2025-08-17T15:49:56","modified_gmt":"2025-08-17T15:49:56","slug":"basic-classes-of-functions-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-classes-of-functions-learn-it-6\/","title":{"raw":"Basic Classes of Functions: Learn It 6","rendered":"Basic Classes of Functions: Learn It 6"},"content":{"raw":"

Transformations of Functions\u00a0<\/h2>\r\n

Understanding how to transform the graph of a function is essential in visualizing mathematical concepts. Transformations<\/strong> include shifting, stretching, or reflecting the graph. Shifting moves the graph up, down, left, or right, stretching alters its width or height, and reflecting flips it over an axis.<\/p>\r\n

Vertical Shift<\/h3>\r\n

Vertical shifts in graphs occur when each point on the graph moves up or down by the same amount. This shift is the result of adding or subtracting a constant to the function's output value [latex]y[\/latex].<\/p>\r\n

For a positive constant [latex]c[\/latex], adding it to a function [latex]f(x)[\/latex] results in [latex]f(x)+c[\/latex], raising the graph up [latex]c[\/latex] units. Conversely, subtracting [latex]c[\/latex] from [latex]f(x)[\/latex] to get [latex]f(x)-c[\/latex] lowers the graph by [latex]c[\/latex] units. These shifts do not affect the shape of the graph; they simply reposition it along the [latex]y[\/latex]-axis.<\/p>\r\n

\r\n

vertical shift<\/h3>\r\n

Vertical shifts do not alter the shape of a function's graph, only its position along the [latex]y[\/latex]-axis. Adding a positive constant lifts the graph upwards, while subtracting it pushes the graph downwards.<\/p>\r\n<\/section>\r\n

\r\n

The graph of the function [latex]f(x)=x^3+4[\/latex] is the graph of [latex]y=x^3[\/latex] shifted up 4 units; the graph of the function [latex]f(x)=x^3-4[\/latex] is the graph of [latex]y=x^3[\/latex] shifted down [latex]4[\/latex] units (Figure 15).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"708\"]\"An Figure 15. (a) For [latex]c>0[\/latex], the graph of [latex]y=f(x)+c[\/latex] is a vertical shift up [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex]. (b) For [latex]c>0[\/latex], the graph of [latex]y=f(x)-c[\/latex] is a vertical shift down [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<\/section>\r\n

Horizontal Shift<\/h3>\r\n

Horizontal shifts in function graphs reflect the influence of adding or subtracting a constant from each input value [latex]x[\/latex].<\/p>\r\n

For a positive constant [latex]c[\/latex], subtracting it from [latex]x[\/latex] to form [latex]f(x-c)[\/latex] shifts the graph to the right by [latex]c[\/latex] units. In contrast, adding [latex]c[\/latex] to [latex]x[\/latex], resulting in\u00a0 [latex]f(x+c)[\/latex], moves the graph to the left by [latex]c[\/latex] units.<\/p>\r\n

\r\n

horizontal shift<\/h3>\r\n

Horizontal shifts alter the position of a function's graph along the [latex]x[\/latex]-axis but do not change its shape. Subtracting a positive constant from the input moves the graph to the right, while adding it shifts the graph to the left<\/p>\r\n<\/section>\r\n

Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let\u2019s look at an example.<\/p>\r\n

\r\n

Consider the function [latex]f(x)=|x+3|[\/latex] and evaluate this function at [latex]x-3.[\/latex] Since [latex]f(x-3)=|x|[\/latex] and [latex]x-3 < x[\/latex], the graph of [latex]f(x)=|x+3|[\/latex] is the graph of [latex]y=|x|[\/latex] shifted left [latex]3[\/latex] units. Similarly, the graph of [latex]f(x)=|x-3|[\/latex] is the graph of [latex]y=|x|[\/latex] shifted right [latex]3[\/latex] units (Figure 16).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"An Figure 16. (a) For [latex]c>0[\/latex], the graph of [latex]y=f(x+c)[\/latex] is a horizontal shift left [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex]. (b) For [latex]c>0[\/latex], the graph of [latex]y=f(x-c)[\/latex] is a horizontal shift right [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<\/section>\r\n

Vertical Scaling (Stretched\/Compressed)<\/h3>\r\n

A vertical scaling of a graph occurs if we multiply all outputs [latex]y[\/latex] of a function by the same positive constant [latex]c[\/latex].<\/p>\r\n

If [latex]c>1[\/latex], the graph of the function [latex]cf(x)[\/latex] appears vertically stretched, as the outputs are proportionally larger than those of the original function [latex]f(x)[\/latex]. If [latex]0 < c <1[\/latex], then the outputs of the function [latex]cf(x)[\/latex] are smaller, so the graph has been compressed, resulting in a graph that is closer to the [latex]x[\/latex]-axis.<\/p>\r\n

\r\n

vertical scaling<\/h3>\r\n

Vertical scaling changes the steepness of a function's graph. Multiplying by a constant greater than [latex]1[\/latex] stretches the graph away from the [latex]x[\/latex]-axis, while multiplying by a constant between [latex]0[\/latex] and [latex]1[\/latex] compresses it towards the [latex]x[\/latex]-axis.<\/p>\r\n<\/section>\r\n

\r\n

The graph of the function [latex]f(x)=3x^2[\/latex] is the graph of [latex]y=x^2[\/latex] stretched vertically by a factor of [latex]3[\/latex], whereas the graph of [latex]f(x)=\\frac{x^2}{3}[\/latex] is the graph of [latex]y=x^2[\/latex] compressed vertically by a factor of [latex]3[\/latex] (Figure 17).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"662\"]\"An Figure 17. (a) If [latex]c>1[\/latex], the graph of [latex]y=cf(x)[\/latex] is a vertical stretch of the graph of [latex]y=f(x)[\/latex]. (b) If [latex]0 < c <1[\/latex], the graph of [latex]y=cf(x)[\/latex] is a vertical compression of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<\/section>\r\n

Horizontal Scaling (Stretched\/Compressed)<\/h3>\r\n


\r\nHorizontal scaling modifies the width of a function\u2019s graph by stretching or compressing it along the [latex]x[\/latex]-axis. This effect is achieved by multiplying the input, [latex]x[\/latex], by a constant [latex]c[\/latex].<\/p>\r\n

When [latex]c>0[\/latex], the function [latex]f(cx)[\/latex] is the graph of [latex]f(x)[\/latex] is compressed, as each input value is effectively scaled down, bringing the points closer together horizontally. If [latex]0 < c <1[\/latex], the function [latex]f(cx)[\/latex] is stretched, because the input values are scaled up, spreading the points further apart on the [latex]x[\/latex]-axis.<\/p>\r\n

\r\n

horizontal scaling<\/h3>\r\n

Horizontal scaling affects the horizontal spread of a function\u2019s graph. Multiplying the input by a constant greater than [latex]1[\/latex] compresses the graph, while a constant between [latex]0[\/latex] and [latex]1[\/latex] stretches it.<\/p>\r\n<\/section>\r\n

\r\n

Consider the function [latex]f(x)=\\sqrt{2x}[\/latex] and evaluate [latex]f[\/latex] at [latex]\\dfrac{x}{2}.[\/latex] Since [latex]f(\\frac{x}{2})=\\sqrt{x}[\/latex], the graph of [latex]f(x)=\\sqrt{2x}[\/latex] is the graph of [latex]y=\\sqrt{x}[\/latex] compressed horizontally. The graph of [latex]y=\\sqrt{\\frac{x}{2}}[\/latex] is a horizontal stretch of the graph of [latex]y=\\sqrt{x}[\/latex] (Figure 18).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"An Figure 18. (a) If [latex]c < 1[\/latex], the graph of [latex]y=f(cx)[\/latex] is a horizontal compression of the graph of [latex]y=f(x)[\/latex]. (b) If [latex]0 < c <1[\/latex], the graph of [latex]y=f(cx)[\/latex] is a horizontal stretch of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<\/section>\r\n

Reflection<\/h3>\r\n

Reflections of a function\u2019s graph across an axis create a mirror image. When we multiply the outputs of a function, [latex]f(x)[\/latex], by [latex]-1[\/latex] we achieve a reflection across the [latex]x[\/latex]-axis, turning every point to its opposite position vertically. Similarly, multiplying the inputs by [latex]-1[\/latex] before applying the function, as in [latex]f(-x)[\/latex], reflects the graph across the y-axis, flipping it horizontally.\u00a0<\/p>\r\n

\r\n

reflections of functions<\/h3>\r\n

Reflecting a function\u2019s graph across an axis is accomplished by multiplying by[latex] -1[\/latex]. To mirror across the [latex]x[\/latex]-axis, multiply the outputs by [latex]-1[\/latex]. To reflect across the [latex]y[\/latex]-axis, multiply the inputs by [latex]-1[\/latex].<\/p>\r\n<\/section>\r\n

\r\n

The graph of [latex]f(x)=\u2212(x^3+1)[\/latex] is the graph of [latex]y=(x^3+1)[\/latex] reflected about the [latex]x[\/latex]-axis. The graph of [latex]f(x)=(\u2212x)^3+1[\/latex] is the graph of [latex]y=x^3+1[\/latex] reflected about the [latex]y[\/latex]-axis (Figure 19).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"An Figure 19. (a) The graph of [latex]y=\u2212f(x)[\/latex] is the graph of [latex]y=f(x)[\/latex] reflected about the [latex]x[\/latex]-axis. (b) The graph of [latex]y=f(\u2212x)[\/latex] is the graph of [latex]y=f(x)[\/latex] reflected about the [latex]y[\/latex]-axis.[\/caption]\r\n<\/section>\r\n

Multiple Transformations<\/h3>\r\n

If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function [latex]f(x)[\/latex], the graph of the related function [latex]y=cf(a(x+b))+d[\/latex] can be obtained from the graph of [latex]y=f(x)[\/latex] by performing the transformations in the following order.<\/p>\r\n

    \r\n\t
  1. Horizontal shift of the graph of [latex]y=f(x)[\/latex]. If [latex]b>0[\/latex], shift left. If [latex]b<0[\/latex], shift right.<\/li>\r\n\t
  2. Horizontal scaling of the graph of [latex]y=f(x+b)[\/latex] by a factor of [latex]|a|[\/latex]. If [latex]a<0[\/latex], reflect the graph about the [latex]y[\/latex]-axis.<\/li>\r\n\t
  3. Vertical scaling of the graph of [latex]y=f(a(x+b))[\/latex] by a factor of [latex]|c|[\/latex]. If [latex]c<0[\/latex], reflect the graph about the [latex]x[\/latex]-axis.<\/li>\r\n\t
  4. Vertical shift of the graph of [latex]y=cf(a(x+b))[\/latex]. If [latex]d>0[\/latex], shift up. If [latex]d<0[\/latex], shift down.<\/li>\r\n<\/ol>\r\n
    \r\n

    We can summarize the different transformations and their related effects on the graph of a function in the following table.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Transformations of Functions<\/caption>\r\n
    Transformation of [latex]f(c>0)[\/latex]<\/strong><\/th>\r\nEffect on the graph of<\/strong>[latex]f[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]f(x)+c[\/latex]<\/td>\r\nVertical shift up [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n
    [latex]f(x)-c[\/latex]<\/td>\r\nVertical shift down [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n
    [latex]f(x+c)[\/latex]<\/td>\r\nShift left by [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n
    [latex]f(x-c)[\/latex]<\/td>\r\nShift right by [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n
    [latex]cf(x)[\/latex]<\/td>\r\nVertical stretch if [latex]c>1[\/latex]; vertical compression if [latex]0 < c < 1[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]f(cx)[\/latex]<\/td>\r\nHorizontal stretch if [latex]0 < c < 1[\/latex]; horizontal compression if [latex]c>1[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\u2212f(x)[\/latex]<\/td>\r\nReflection about the [latex]x[\/latex]-axis<\/td>\r\n<\/tr>\r\n
    [latex]f(\u2212x)[\/latex]<\/td>\r\nReflection about the [latex]y[\/latex]-axis<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n
    \r\n

    Describe how the function [latex]f(x)=\u2212(x+1)^2-4[\/latex] can be graphed using the graph of [latex]y=x^2[\/latex] and a sequence of transformations.<\/p>\r\n

    [reveal-answer q=\"fs-id1170573581275\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"fs-id1170573581275\"]<\/p>\r\n\r\nTo graph the function [latex]f(x)=\u2212(x+1)^2-4[\/latex] using transformations, start with the base function [latex]g(x)=x^2[\/latex] and follow these steps:\r\n\r\n

      \r\n\t
    1. Horizontal Shift<\/strong>: The term [latex](x+1)[\/latex] within [latex](x+1)^2[\/latex] indicates a horizontal shift of the graph of [latex]g(x)[\/latex] one unit to the left.\u00a0<\/li>\r\n\t
    2. Vertical Shift<\/strong>: The [latex]\u22124[\/latex] at the end of [latex](x+1)^2\u22124[\/latex] indicates a vertical shift downward by [latex]4[\/latex] units. This is a result of subtracting [latex]4[\/latex] from the entire squared term.<\/li>\r\n\t
    3. Reflection<\/strong>: The negative sign in front of the function indicates that the graph will be a reflection of [latex]g(x)=x^2[\/latex] across the [latex]x[\/latex]-axis. This means that the parabola, which normally opens upwards, will now open downwards.<\/li>\r\n<\/ol>\r\n\r\nTo graph [latex]f(x)[\/latex]:\r\n\r\n
        \r\n\t
      • Begin with the graph of [latex]g(x)=x^2[\/latex], which is a parabola with its vertex at the origin [latex](0,0)[\/latex].<\/li>\r\n\t
      • Reflecting it across the x-axis due to the negative sign, which will flip the parabola to open downwards.<\/li>\r\n\t
      • Move this graph one unit left to accommodate the [latex]+1[\/latex] within the squared term, shifting the vertex to [latex](\u22121,0)[\/latex].<\/li>\r\n\t
      • Then, shift the graph down four units for the [latex]\u22124[\/latex], placing the vertex at [latex](\u22121,\u22124)[\/latex].<\/li>\r\n<\/ul>\r\n\r\nThe transformed graph of [latex]f(x)[\/latex] will be a downward-opening parabola with its vertex at [latex](\u22121,\u22124)[\/latex].\r\n\r\n

        [\/hidden-answer]<\/p>\r\n<\/section>\r\n

        \r\n

        It is beneficial when working with transformations to remember the basic toolkit functions. These will be your starting points when trying to identify how the function has been transformed.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        Toolkit Functions<\/th>\r\n<\/tr>\r\n
        Name<\/th>\r\nFunction<\/th>\r\nGraph<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
        Constant<\/td>\r\n[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"517\"]\"Graph Graph of a constant function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Identity\/Linear<\/td>\r\n[latex]f\\left(x\\right)=x[\/latex]<\/td>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"517\"]\"Graph Graph of a linear function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Absolute value<\/td>\r\n[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"517\"]\"Graph Graph of an absolute function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Quadratic<\/td>\r\n[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n\r\n[caption id=\"attachment_16077\" align=\"alignnone\" width=\"567\"]\"Graph<\/a> Graph of a quadratic function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Cubic<\/td>\r\n[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"517\"]\"Graph Graph of a cubic function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Reciprocal<\/td>\r\n[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"517\"]\"Graph Graph of a reciprocal function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Reciprocal squared<\/td>\r\n[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n\r\n[caption id=\"attachment_16079\" align=\"alignnone\" width=\"562\"]\"Graph<\/a> Graph of a reciprocal squared function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Square root<\/td>\r\n[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\r\n\r\n[caption id=\"attachment_16071\" align=\"alignnone\" width=\"565\"]\"Graph<\/a> Graph of a square root function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n
        Cube root<\/td>\r\n[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"517\"]\"Graph Graph of a cube root function[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n
        \r\n

        For each of the following functions, sketch a graph by using a sequence of transformations of a toolkit function.<\/p>\r\n

          \r\n\t
        1. [latex]f(x)=\u2212|x+2|-3[\/latex]<\/li>\r\n\t
        2. [latex]f(x)=3\\sqrt{\u2212x}+1[\/latex]<\/li>\r\n<\/ol>\r\n

          [reveal-answer q=\"fs-id1170573580983\"]Show Solution[\/reveal-answer]
          \r\n[hidden-answer a=\"fs-id1170573580983\"]<\/p>\r\n

            \r\n\t
          1. Starting with the graph of [latex]y=|x|[\/latex], shift [latex]2[\/latex] units to the left, reflect about the [latex]x[\/latex]-axis, and then shift down [latex]3 [\/latex] units.\r\n[caption id=\"\" align=\"aligncenter\" width=\"479\"]\"An Figure 20. The function [latex]f(x)=\u2212|x+2|-3[\/latex] can be viewed as a sequence of three transformations of the function [latex]y=|x|[\/latex].[\/caption]\r\n<\/li>\r\n\t
          2. Starting with the graph of [latex]y=\\sqrt{x}[\/latex], reflect about the [latex]y[\/latex]-axis, stretch the graph vertically by a factor of [latex]3[\/latex], and move up [latex]1[\/latex] unit.\r\n[caption id=\"\" align=\"aligncenter\" width=\"479\"]\"An Figure 21. The function [latex]f(x)=3\\sqrt{\u2212x}+1[\/latex] can be viewed as a sequence of three transformations of the function [latex]y=\\sqrt{x}[\/latex].[\/caption]\r\n<\/li>\r\n<\/ol>\r\n\r\nWatch the following video to see the worked solution to this example.