Graphing a Function In Exercises 43-50, sketch the graph of the function. \( \begin{array}{ll}\text { 43. } f(x)=x^{2} & \text { 44. } f(x)=x^{3} \\ \text { 45. } f(x)=\sqrt{x} & \text { 46. } f(x)=\frac{1}{x} \\ \text { 47. } g(x)=\llbracket x \rrbracket-2 & \\ \text { 48. } g(x)=\llbracket x+4 \rrbracket & x \geq-1 \\ \text { 49. } f(x)=\left\{\begin{array}{ll}5 x-3, & x \geq-4 x+5, \\ -4<-1\end{array}\right. \\ \text { 50. } f(x)=\left\{\begin{array}{ll}2 x+1, & x \leq 2 \\ x^{2}+1, & x>2\end{array}\right.\end{array} \)

Certainly! Let's go through each of the functions from exercises **43** to **50** and discuss how to sketch their graphs, highlighting key features such as shape, intercepts, symmetry, asymptotes, and any special behaviors. --- ## **43. \( f(x) = x^{2} \)** ### **Type of Function:** - **Quadratic Function**. ### **Graph Shape:** - **Parabola** opening **upwards**. ### **Key Features:** - **Vertex:** At the origin \((0, 0)\). - **Axis of Symmetry:** The y-axis (\(x = 0\)). - **Intercepts:** - **Y-intercept:** \((0, 0)\). - **X-intercepts:** \((0, 0)\) (double root). - **Symmetry:** **Even Function** (symmetric about the y-axis). ### **Steps to Sketch:** 1. Plot the vertex at \((0, 0)\). 2. Choose several x-values (both positive and negative) and compute corresponding y-values. 3. Plot points like \((1, 1)\), \((-1, 1)\), \((2, 4)\), etc. 4. Draw a smooth, symmetrical U-shaped curve passing through these points. ### **Graph:** ``` | 4 | * | * * 2 | * * | * * ------0 |--------------*-------- | * * -2 | * * | * * -4 | * | ``` --- ## **44. \( f(x) = x^{3} \)** ### **Type of Function:** - **Cubic Function**. ### **Graph Shape:** - An **S-shaped** curve. ### **Key Features:** - **Origin:** Passes through \((0, 0)\). - **Symmetry:** **Odd Function** (symmetric about the origin). - **Intercepts:** - **Y-intercept:** \((0, 0)\). - **X-intercept:** \((0, 0)\). - **Behavior:** - As \(x \to \infty\), \(f(x) \to \infty\). - As \(x \to -\infty\), \(f(x) \to -\infty\). ### **Steps to Sketch:** 1. Plot the origin \((0, 0)\). 2. Calculate points for positive and negative x-values, such as \((1, 1)\), \((-1, -1)\), \((2, 8)\), \((-2, -8)\). 3. Draw the S-shaped curve passing through these points, extending to infinity in both directions. ### **Graph:** ``` | 8 | * | * 4 | * | * ------0 |--*---------*--------- | * -4 | * | * -8 | * | ``` --- ## **45. \( f(x) = \sqrt{x} \)** ### **Type of Function:** - **Square Root Function**. ### **Graph Shape:** - A **half-parabola** starting from the origin and increasing to the right. ### **Key Features:** - **Domain:** \(x \geq 0\). - **Range:** \(y \geq 0\). - **Intercepts:** - **Y-intercept:** \((0, 0)\). - **No X-intercept** (other than at the origin). - **Shape:** Rises rapidly near the origin and grows more slowly as \(x\) increases. ### **Steps to Sketch:** 1. Plot the origin \((0, 0)\). 2. Choose values of \(x\) such as \(1, 4, 9, 16\) and plot \((1,1)\), \((4,2)\), \((9,3)\), \((16,4)\). 3. Draw a curve starting at the origin, increasing gradually to the right. ### **Graph:** ``` | 4 | * | 3 | * | 2 | * | 1 | * ------0 |*----------------------- | ``` --- ## **46. \( f(x) = \frac{1}{x} \)** ### **Type of Function:** - **Rational Function** (Hyperbola). ### **Graph Shape:** - Two separate branches: one in the first quadrant and one in the third quadrant. ### **Key Features:** - **Domain:** \(x \neq 0\). - **Range:** \(y \neq 0\). - **Asymptotes:** - **Vertical Asymptote:** \(x = 0\). - **Horizontal Asymptote:** \(y = 0\). - **Intercepts:** - **No X-intercept** or Y-intercept. - **Symmetry:** **Odd Function** (symmetric about the origin). ### **Steps to Sketch:** 1. Draw the asymptotes: y-axis and x-axis. 2. Plot points for \(x > 0\), such as \((1,1)\), \((2,0.5)\), \((0.5,2)\). 3. Plot points for \(x < 0\), such as \((-1,-1)\), \((-2,-0.5)\), \((-0.5,-2)\). 4. Draw the hyperbola branches approaching the asymptotes. ### **Graph:** ``` | * | * | ------0 |------------*------------ | | * | * | ``` *Note: The asterisks (*) represent points approaching the asymptotes.* --- ## **47. \( g(x) = \lfloor x \rfloor - 2 \)** ### **Type of Function:** - **Step Function** (Floor Function). ### **Graph Shape:** - **Discrete** steps, each step is 1 unit high and 1 unit wide. ### **Key Features:** - **Domain:** All real numbers. - **Range:** All integers minus 2. - **Behavior:** - For any \(x\), \(g(x)\) is the greatest integer less than or equal to \(x\) minus 2. - Left-continuous: includes the left endpoint, excludes the right. ### **Steps to Sketch:** 1. For each integer \(n\), the function \(g(x)\) is constant at \(n - 2\) for \(x\) in \([n, n+1)\). 2. Plot horizontal lines at each integer value minus 2, from each integer \(x\) to the next, excluding the right endpoint. 3. Indicate open circles at the right endpoints and closed circles at the left. ### **Example:** - For \(x\) in \([0,1)\), \(g(x) = \lfloor 0 \rfloor - 2 = -2\). - For \(x\) in \([1,2)\), \(g(x) = 1 - 2 = -1\), etc. ### **Graph:** ``` y | 1 | 0 | -1|---------●--------- -2|---------●----●---- -3|----●----●----●---- 0 1 2 3 x ``` *Note: Each "●" represents a closed circle at the left endpoint of each step, with horizontal lines extending to the next integer.* --- ## **48. \( g(x) = \lfloor x + 4 \rfloor \), \( x \geq -1 \)** ### **Type of Function:** - **Step Function** (Floor Function) with a horizontal shift. ### **Graph Shape:** - Similar to a standard floor function but shifted left by 4 units. - Defined only for \(x \geq -1\). ### **Key Features:** - **Domain:** \(x \geq -1\). - **Range:** All integers starting from \(\lfloor -1 + 4 \rfloor = 3\). - **Behavior:** - For any \(x\), \(g(x) = \lfloor x + 4 \rfloor\). - Steps occur at integer values of \(x + 4\), i.e., \(x = n - 4\) where \(n\) is an integer. - **Symmetry:** None. ### **Steps to Sketch:** 1. Since \(x \geq -1\), the smallest value inside the floor is \(-1 + 4 = 3\). 2. For \(x \geq -1\) up to \(x < 0\), \(g(x) = 3\). 3. From \(x = 0\) to \(x <1\), \(g(x) = 4\); from \(x =1\) to \(x <2\), \(g(x)=5\), and so on. 4. Plot horizontal steps starting at \(x = -1\), each step one unit wide, jumping up by 1 at each integer \(x\). ### **Graph:** ``` y | 7 | --------- 6 | -- -- 5 | -- -- 4 | -- -- 3 |----●--------------------- -1 0 1 2 x ``` *Note: The steps start at \(x = -1\) with \(g(x) = 3\), increasing by 1 at each integer \(x\). The "●" marks the start at \(x = -1\).* --- ## **49. \( f(x) = \begin{cases} 5x - 3, & x \geq -1 \\ x + 5, & -4 < x < -1 \end{cases} \)** *Assumption: The original problem likely intended the piecewise function to be defined with continuity at \(x = -1\).* ### **Type of Function:** - **Piecewise Linear Function**. ### **Graph Shape:** - Two straight line segments with different slopes. ### **Key Features:** - **Domain:** - \(x \geq -1\): \(5x - 3\). - \(-4 < x < -1\): \(x + 5\). - **Intercepts:** - **For \(5x - 3\):** - Y-intercept: \((0, -3)\). - Slope: 5 (steep upward). - **For \(x + 5\):** - Y-intercept: \((0, 5)\). - Slope: 1 (less steep). - **Continuity:** Check at \(x = -1\). - \(f(-1^-)= -1 +5=4\). - \(f(-1^+)=5*(-1)-3=-8\). - **Discontinuity:** Jump from 4 to -8 at \(x = -1\). ### **Steps to Sketch:** 1. **For \(-4 < x < -1\):** - Plot points like \((-3, 2)\), \((-2, 3)\), approaching \(x = -1\) with \(y = 4\). - Draw a line with slope 1 up to \(x = -1\), open circle at \((-1,4)\). 2. **For \(x \geq -1\):** - Start at \(x = -1\) with \(y = -8\) (closed circle at \((-1,-8)\)). - Plot points like \((0,-3)\), \((1,2)\), etc., with slope 5. ### **Graph:** ``` y | 5 | * 4 | o 3 | * 2 | * 1 | 0 | -1 | -2 | -3 |----------●---------- -4 | (-1,-8) -4 -3 -2 -1 0 1 x ``` *Explanation: The segment \(-4 < x < -1\) is a line \(y = x + 5\) ending at \((-1,4)\) (open circle), and for \(x \geq -1\), the line \(y = 5x - 3\) starts at \((-1,-8)\) (closed circle) and continues upwards.* --- ## **50. \( f(x) = \begin{cases} 2x + 1, & x \leq 2 \\ x^{2} + 1, & x > 2 \end{cases} \)** ### **Type of Function:** - **Piecewise Function** combining Linear and Quadratic functions. ### **Graph Shape:** - A straight line for \(x \leq 2\) and a parabola for \(x > 2\). ### **Key Features:** - **Domain:** - \(x \leq 2\): \(2x + 1\). - \(x > 2\): \(x^{2} + 1\). - **Continuity at \(x = 2\):** - Left-hand limit: \(2(2) +1=5\). - Right-hand value: \(2^{2}+1=5\). - **Continuous** at \(x = 2\). - **Intercepts:** - **For \(2x + 1\):** - Y-intercept: \((0,1)\). - Slope: 2. - **For \(x^{2} + 1\):** - Vertex: \((0,1)\) shifted upwards by 1. - Opening upwards. - **Behavior:** - For \(x \leq 2\), a line increasing with slope 2. - For \(x > 2\), a parabola opening upwards starting at \((2,5)\). ### **Steps to Sketch:** 1. **For \(x \leq 2\):** - Plot the line \(y = 2x + 1\), passing through \((0,1)\), and \((2,5)\). - Include the point \((2,5)\). 2. **For \(x > 2\):** - Plot the parabola \(y = x^{2} + 1\), starting at \((2,5)\). - Plot points like \((3,10)\), \((4,17)\), etc. 3. Ensure the graph is continuous at \(x = 2\). ### **Graph:** ``` y | 17| * 16| 15| 10| * 5 |---------●---------* 4 | 3 | 2 | 1 |●------------------- 0 1 2 3 4 x ``` *Explanation: The line \(y = 2x +1\) is drawn up to \(x = 2\), where it meets smoothly with the parabola \(y = x^{2} +1\) starting at \((2,5)\). The "●" at \(x = 2\) ensures continuity.* --- ## **Summary of Graphing Techniques:** 1. **Identify the Type of Function:** Understanding whether it's linear, quadratic, cubic, rational, step, or piecewise is crucial. 2. **Determine Domain and Range:** Establish where the function is defined and the possible output values. 3. **Find Intercepts:** Calculate where the graph crosses the x-axis and y-axis. 4. **Check for Symmetry:** Determine if the function is even, odd, or has no symmetry. 5. **Locate Asymptotes (if any):** For functions like \(1/x\), identify vertical and horizontal asymptotes. 6. **Plot Key Points:** Select important x-values and compute corresponding y-values to plot on the graph. 7. **Draw the Graph:** Connect the plotted points smoothly, respecting the function's continuity and behavior. By following these steps for each function, you can accurately sketch their graphs and understand their behaviors.