2.5.2: Homework

Math 375: Pre-Calculus · LibreTexts

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    From LibreTexts, under CC BY-SA, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Reading Questions

    1. How does the transformation \(g(x) = f(x) + 5\) affect the graph of \(f(x)\)?
    2. If the graph of \(f(x)\) is shifted 3 units to the left, what is the equation of the new function in terms of \(f(x)\)?
    3. What is the difference in the graphical effect of \(g(x) = -f(x)\) versus \(h(x) = f(-x)\)?
    4. If \(f(x) = x^2\), describe the transformation that results in \(g(x) = 3x^2\). Is this a stretch or a compression? In which direction?
    5. If \(f(x) = \sqrt{x}\), describe the transformation that results in \(g(x) = \sqrt{2x}\). Is this a stretch or a compression? In which direction? By what factor?
    6. When transforming a function using the form \(y = af(bx+p)+k\), why is it recommended to first factor out \(b\) from the term \((bx+p)\)?
    7. Explain why \(f(x-2)\) shifts the graph of \(f(x)\) to the right, even though a subtraction is involved.
    8. If you are given the graph of \(y=f(x)\) and want to graph \(y = 2f(x) - 3\), describe the sequence of transformations you would apply.
    9. What is the domain and range of \(g(t) = \sqrt{-t}\) if the original function is \(s(t) = \sqrt{t}\) with domain \([0, \infty)\) and range \([0, \infty)\)?
    10. Consider the function \(k(t) = -2^{-t} + 1\). Identify the base function and list the transformations applied to it in the correct order to obtain \(k(t)\).
    11. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?
    12. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?
    13. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?
    14. When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the \( x \)-axis from a reflection with respect to the \( y \)-axis?

    Homework

    Algebraic

    1. Write a formula for the function obtained when the graph of \(f(x)=\sqrt{ x }\) is shifted up 1 unit and to the left 2 units.
    2. Write a formula for the function obtained when the graph of \(f(x)=| x |\) is shifted down 3 units and to the right 1 unit.
    3. Write a formula for the function obtained when the graph of \(f(x)= \frac{1}{x}\) is shifted down 4 units and to the right 3 units.
    4. Write a formula for the function obtained when the graph of \(f(x)= \frac{1}{x^2}\) is shifted up 2 units and to the left 4 units.

    For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\).

    1. \(y=f(x−49)\)
    2. \(y=f(x+43)\)
    3. \(y=f(x+3)\)
    4. \(y=f(x−4)\)
    5. \(y=f(x)+5\)
    6. \(y=f(x)+8\)
    7. \(y=f(x)−2\)
    8. \(y=f(x)−7\)
    9. \(y=f(x−2)+3\)
    10. \(y=f(x+4)−1\)

    For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

    1. \(f(x)=4 (x+1)^2 −5\)
    2. \(g(x)=5 (x+3)^2 −2\)
    3. \(a(x)=\sqrt{ −x+4 }\)
    4. \(k(x)=−3\sqrt{ x } −1\)

    Graphical

    For the following exercises, use the graph of \(f(x)= 2^x\) shown in Figure \( \PageIndex{ 1 } \) to sketch a graph of each transformation of \(f(x)\).

    Figure \( \PageIndex{ 1 } \)
     
    Graph of f(x).
    1. \(g(x)= 2^x +1\)
    2. \(h(x)= 2^x −3\)
    3. \(w(x)= 2^{x−1}\)

    For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

    1. \(f(t)= (t+1)^2 −3\)
    2. \(h(x)=|x−1|+4\)
    3. \(k(x)= (x−2)^3 −1\)
    4. \(m(t)=3+\sqrt{ t+2 }\)

    Numeric

    1. Tabular representations for the functions \( f \), \( g \), and \(h\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\).
    \(x\) −2 −1 0 1 2
    \(f(x)\) −2 −1 −3 1 2
    \(x\) −1 0 1 2 3
    \(g(x)\) −2 −1 −3 1 2
    \(x\) −2 −1 0 1 2
    \(h(x)\) −1 0 −2 2 3
    1. Tabular representations for the functions \( f \), \( g \), and \(h\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\).
    \(x\) −2 −1 0 1 2
    \(f(x)\) −1 −3 4 2 1
    \(x\) −3 −2 −1 0 1
    \(g(x)\) −1 −3 4 2 1
    \(x\) −2 −1 0 1 2
    \(h(x)\) −2 −4 3 1 0

    For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

    1.  
      Graph of an absolute function.
    2.  
      Graph of a parabola.
    3.  
      Graph of a square root function.
    4.  
      Graph of an absolute function.
    5.  
      Graph of a parabola
    6.  
      Graph of a square root function.
    7.  
      Graph of an absolute function.
    8.  
      Graph of a square root function.

    For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

    1.  
      Graph of a square root function.
    2.  
      Graph of a square root function.

    For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

    1.  
      Graph of a parabola.
    2.  
      Graph of a cubic function.
    3.  
      Graph of a square root function.
    4.  
      Graph of an absolute function.

    For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\).

    1. \(g(x)=−f(x)\)
    2. \(g(x)=f(−x)\)
    3. \(g(x)=4f(x)\)
    4. \(g(x)=6f(x)\)
    5. \(g(x)=f(5x)\)
    6. \(g(x)=f(2x)\)
    7. \(g(x)=f\left( \frac{1}{3} x \right)\)
    8. \(g(x)=f\left( \frac{1}{5} x \right)\)
    9. \(g(x)=3f( −x )\)
    10. \(g(x)=−f(3x)\)

    For the following exercises, write a formula for the function \(g\) that results when the graph of a given toolkit function is transformed as described.

    1. The graph of \(f(x)=|x|\) is reflected over the \(y\)-axis and horizontally compressed by a factor of \(\frac{1}{4}\) .
    2. The graph of \(f(x)=\sqrt{ x }\) is reflected over the \(x\)-axis and horizontally stretched by a factor of 2.
    3. The graph of \(f(x)= \frac{1}{x^2}\) is vertically compressed by a factor of \(\frac{1}{3} \), then shifted to the left 2 units and down 3 units.
    4. The graph of \(f(x)= \frac{1}{x}\) is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.
    5. The graph of \(f(x)= x^2\) is vertically compressed by a factor of \(\frac{1}{2} \), then shifted to the right 5 units and up 1 unit.
    6. The graph of \(f(x)= x^2\) is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

    For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

    1. \(g(x)=4 (x+1)^2 −5\)
    2. \(g(x)=5 (x+3)^2 −2\)
    3. \(h(x)=−2|x−4|+3\)
    4. \(k(x)=−3\sqrt{ x } −1\)
    5. \(m(x)= \frac{1}{2} x^3\)
    6. \(n(x)= \frac{1}{3} |x−2|\)
    7. \(p( x )= \left( \frac{1}{3} x \right)^3 −3\)
    8. \(q( x )= \left( \frac{1}{4} x \right)^3 +1\)
    9. \(a(x)=\sqrt{ −x+4 }\)

    For the following exercises, use the graph in Figure \( \PageIndex{ 2 } \) to sketch the given transformations.

    Figure \( \PageIndex{ 2 } \)
     
    Graph of a polynomial.
    1. \(g(x)=f(x)−2\)
    2. \(g(x)=−f(x)\)
    3. \(g(x)=f(x+1)\)
    4. \( g(x)=f(x−2)\)

    2.5.2: Homework is shared under a CC BY-SA license and was authored, remixed, and/or curated by LibreTexts.