Let's go through **each equation** step by step and describe what each surface is and how you would graph it. --- ### 1. \( x^2 + z^2 = 9 \) **Step 1:** Recognize the form. This is the equation of a **circle** in the \(xz\)-plane with center at \((0,0)\) and radius 3. **Step 2:** Since \(y\) does not appear, this is actually a **cylinder** extending infinitely along the \(y\)-axis. **Graphing:** - Draw the \(xz\)-plane. - Draw a circle centered at the origin with radius 3. - Indicate that this circle is extruded along the \(y\)-axis (cylinder). --- ### 2. \( 4y + 3z = 12 \) **Step 1:** Rewrite as \( y = -\frac{3}{4}z + 3 \). **Step 2:** This is the equation of a **plane** in three dimensions. **Graphing:** - The intercepts are: - When \(y=0\): \(z=4\) - When \(z=0\): \(y=3\) - Plot these intercepts and draw a plane through them. --- ### 3. \( \frac{x^2}{16} + \frac{y^2}{25} + \frac{z^2}{9} = 1 \) **Step 1:** Recognize the form. This is the equation of an **ellipsoid** centered at the origin. **Step 2:** The axes are: - Along \(x\): length 4 (\(\sqrt{16}\)) - Along \(y\): length 5 (\(\sqrt{25}\)) - Along \(z\): length 3 (\(\sqrt{9}\)) **Graphing:** - Draw an ellipse in each coordinate plane with the correct axis lengths. - Sketch an ellipsoid shape. --- ### 4. \( x^2 + y^2 = z \) **Step 1:** Rearranged: \(z = x^2 + y^2\) **Step 2:** This is a **paraboloid** opening along the positive \(z\)-axis. **Graphing:** - At \(z=0\), point is at the origin. - For increasing \(z\), circles of radius \(\sqrt{z}\). - Draw a bowl-shaped surface opening upwards. --- ### 5. \( \frac{x^2}{4} - \frac{y^2}{16} + \frac{z^2}{9} = 1 \) **Step 1:** Recognize the form. This is a **hyperboloid of one sheet** because one variable is negative and the other two are positive. **Step 2:** The axes are: - Along \(x\): length 2 (\(\sqrt{4}\)) - Along \(y\): length 4 (\(\sqrt{16}\)), but negative sign means hyperbola - Along \(z\): length 3 (\(\sqrt{9}\)) **Graphing:** - Draw hyperbolic cross-sections in the \(xy\)-plane. - Elliptical cross-sections in \(xz\) and \(yz\)-planes. - Shape looks like a "saddle" or "hourglass". --- ### 6. \( \frac{z^2}{9} - \frac{x^2}{4} - \frac{y^2}{4} = 1 \) **Step 1:** Recognize the form. This is a **hyperboloid of two sheets** because two variables are negative and one is positive. **Step 2:** The axes are: - Along \(z\): length 3 (\(\sqrt{9}\)) - Along \(x\): length 2 (\(\sqrt{4}\)), negative sign - Along \(y\): length 2 (\(\sqrt{4}\)), negative sign **Graphing:** - For \(|z| > 3\), real solutions exist. - Shape: two separate "bowls" opening along the \(z\)-axis, above and below. --- ## **Summary Table** | # | Equation | Surface Type | Key Features | |---|----------|-------------|--------------| | 1 | \(x^2 + z^2 = 9\) | Cylinder | Circle in \(xz\)-plane, extends in \(y\) | | 2 | \(4y + 3z = 12\) | Plane | Linear, intercepts at \(y=3\), \(z=4\) | | 3 | \(\frac{x^2}{16} + \frac{y^2}{25} + \frac{z^2}{9} = 1\) | Ellipsoid | Center at origin, axes: 4, 5, 3 | | 4 | \(x^2 + y^2 = z\) | Paraboloid | Opens along \(z\)-axis | | 5 | \(\frac{x^2}{4} - \frac{y^2}{16} + \frac{z^2}{9} = 1\) | Hyperboloid (1-sheet) | "Hourglass" shape | | 6 | \(\frac{z^2}{9} - \frac{x^2}{4} - \frac{y^2}{4} = 1\) | Hyperboloid (2-sheets) | Two bowls, opens along \(z\) | If you need actual hand-drawn graphs, let me know!

Certainly! Let's go through each equation carefully with detailed explanations, calculations, and how you would graph each surface by hand. --- ### 1. \( x^2 + z^2 = 9 \) **Type:** Cylinder **Step-by-step explanation:** - **Identify the form:** The equation resembles a circle in the \(xz\)-plane: \[ x^2 + z^2 = r^2 \] with \(r^2 = 9\), so \(r=3\). - **Interpretation:** Since \(y\) does not appear in the equation, this describes a **cylinder** extending infinitely along the \(y\)-axis. - **Graphing process:** - Draw the \(xz\)-plane. - Plot a circle centered at \((0,0)\) with radius 3: Points like \((3,0)\), \((-3,0)\), \((0,3)\), \((0,-3)\). - Extend this circle infinitely along the \(y\)-direction: draw lines parallel to the \(y\)-axis passing through the circle's boundary points. - The resulting surface is a **cylinder** with a circular cross-section. --- ### 2. \( 4y + 3z = 12 \) **Type:** Plane **Step-by-step explanation:** - **Rewrite the equation:** Solve for \(y\): \[ 4y = 12 - 3z \implies y = -\frac{3}{4}z + 3 \] - **Intercepts:** - When \(z=0\), \[ y = -\frac{3}{4} \times 0 + 3 = 3 \] So point: \((x, y, z) = (\text{any } x, 3, 0)\). - When \(y=0\), \[ 0 = -\frac{3}{4}z + 3 \implies -\frac{3}{4}z = -3 \implies z=4 \] So point: \((x, 0, 4)\). - **Graphing process:** - Plot these intercept points: \((x, y, z) = (x, 3, 0)\) and \((x, 0, 4)\). - Draw the plane passing through these points. - The plane cuts through the space, extending infinitely in all directions. --- ### 3. \( \frac{x^2}{16} + \frac{y^2}{25} + \frac{z^2}{9} = 1 \) **Type:** Ellipsoid **Step-by-step explanation:** - **Identify axes lengths:** - Along \(x\): \[ \frac{x^2}{16} = 1 \implies x^2 = 16 \implies x = \pm 4 \] - Along \(y\): \[ y^2/25=1 \implies y = \pm 5 \] - Along \(z\): \[ z^2/9=1 \implies z = \pm 3 \] - **Graphing process:** - Draw an ellipse in the \(xy\)-plane at \(z=0\) with axes 4 (along \(x\)) and 5 (along \(y\)). - Similarly, draw ellipses in the \(xz\)- and \(yz\)-planes with the appropriate axes. - Sketch a smooth, oval-shaped ellipsoid centered at the origin, with the axes lengths as above. --- ### 4. \( x^2 + y^2 = z \) **Type:** Paraboloid **Step-by-step explanation:** - **Rearranged as:** \[ z = x^2 + y^2 \] - **Interpretation:** For each fixed \(z\), the cross-section in the \(xy\)-plane is a circle of radius \(\sqrt{z}\). - When \(z=0\), point at the origin. - For \(z>0\), circles grow larger as \(z\) increases. - **Graphing process:** - Draw the \(xy\)-plane and mark the circle at \(z=0\): just the origin. - For increasing \(z\), draw circles with radius \(\sqrt{z}\). - Connect these in a smooth, bowl-shaped surface opening upward. --- ### 5. \( \frac{x^2}{4} - \frac{y^2}{16} + \frac{z^2}{9} = 1 \) **Type:** Hyperboloid of one sheet **Step-by-step explanation:** - **Identify axes and shape:** - \(x^2/4\): \(x\)-axis length \(\pm 2\) - \(- y^2/16\): negative term, indicating a hyperbolic shape in \(y\) - \(+ z^2/9\): positive, elongated along \(z\) - **Key features:** - Cross-sections at constant \(z\): \[ \frac{x^2}{4} - \frac{y^2}{16} = 1 - \frac{z^2}{9} \] For \(|z| < 3\), the right side is positive, giving hyperbolas. - As \(z\) increases, the hyperbolas widen. - **Graphing process:** - Draw hyperbolas in the \(xy\)-plane at various \(z\)-values. - The surface looks like a "saddle" or "hourglass" shape with a central "waist" at \(z=0\). --- ### 6. \( \frac{z^2}{9} - \frac{x^2}{4} - \frac{y^2}{4} = 1 \) **Type:** Hyperboloid of two sheets **Step-by-step explanation:** - **Identify axes:** - \(z^2/9\): positive, opening along \(z\)-axis. - \(- x^2/4\), \(- y^2/4\): negative, indicating the surface exists only for certain \(z\). - **Key features:** - For \(|z| > 3\), the equation holds because the left side is positive and equals 1. - The surface consists of two separate "bowls" or sheets, above and below the plane \(z=0\). - **Graphing process:** - For \(z > 3\), draw the cross-section in the \(xy\)-plane: \[ - \frac{x^2}{4} - \frac{y^2}{4} = 1 - \frac{z^2}{9} \] which is valid only when the RHS is positive, i.e., \(|z|>3\). - Sketch two separate sheets opening along \(z\), symmetric about the plane \(z=0\). --- ## **Summary of Surfaces:** | # | Equation | Surface Type | Key Features | |---|------------------------------------------------------------|------------------------------|------------------------------| | 1 | \(x^2 + z^2 = 9\) | Circular cylinder | Extends infinitely along \(y\) | | 2 | \(4y + 3z = 12\) | Plane | Intercepts at \(y=3\), \(z=4\) | | 3 | \(\frac{x^2}{16} + \frac{y^2}{25} + \frac{z^2}{9} = 1\) | Ellipsoid | Centered at origin with axes 4,5,3 | | 4 | \(z = x^2 + y^2\) | Paraboloid | Bowl shape opening upward | | 5 | \(\frac{x^2}{4} - \frac{y^2}{16} + \frac{z^2}{9} = 1\) | Hyperboloid (1-sheet) | "Hourglass" shape | | 6 | \(\frac{z^2}{9} - \frac{x^2}{4} - \frac{y^2}{4} = 1\) | Hyperboloid (2-sheets) | Two separate "bowls" along \(z\) | --- **Final note:** When graphing by hand, start with the basic shapes: - Draw the intercepts and cross-sections in each principal plane. - Sketch the general surface shape, ensuring symmetry and correct axes lengths. - Use a ruler for straight lines (planes) and smooth freehand for curved surfaces. Feel free to ask if you'd like a visual sketch guide!

Certainly! Here are the final answers for each equation: 1. **\(x^2 + z^2 = 9\)** - Surface: **Cylinder** extending infinitely along the \(y\)-axis, with a circular cross-section of radius 3 in the \(xz\)-plane. 2. **\(4y + 3z = 12\)** - Surface: **Plane** intersecting the axes at \(y=3\) and \(z=4\). 3. **\(\frac{x^2}{16} + \frac{y^2}{25} + \frac{z^2}{9} = 1\)** - Surface: **Ellipsoid** centered at the origin with semi-axes 4 (x), 5 (y), and 3 (z). 4. **\(x^2 + y^2 = z\)** - Surface: **Paraboloid** opening upward along the \(z\)-axis. 5. **\(\frac{x^2}{4} - \frac{y^2}{16} + \frac{z^2}{9} = 1\)** - Surface: **Hyperboloid of one sheet** (hourglass shape). 6. **\(\frac{z^2}{9} - \frac{x^2}{4} - \frac{y^2}{4} = 1\)** - Surface: **Hyperboloid of two sheets**, with two separate "bowls" opening along \(\pm z\). These are the types and key features of each quadratic surface.