Graphs of Parabolas - Vertex Form | PDF - Free Printable

Let's solve each of these parabola problems step by step. The equations are given in vertex form, which is:

$$
y = a(x - h)^2 + k
$$

Where:
- $(h, k)$ is the vertex,
- The axis of symmetry is $x = h$,
- If $a > 0$, the parabola opens upward (minimum value at vertex),
- If $a < 0$, it opens downward (maximum value at vertex),
- The y-intercept is found by setting $x = 0$,
- The x-intercepts are found by setting $y = 0$ and solving for $x$.

We'll go through each problem one by one.

---

1) $ y = (x - 5)^2 - 4 $

#### Vertex:
From $ y = (x - 5)^2 - 4 $, we see:
- $ h = 5 $, $ k = -4 $
- Vertex: $ (5, -4) $

#### Axis of Symmetry:
- $ x = 5 $

#### Direction of Opening:
- Coefficient $ a = 1 > 0 $ → Opens upward

#### Min/Max Value:
- Since it opens upward, has a minimum value at vertex: $ y = -4 $

#### Y-intercept:
Set $ x = 0 $:
$$
y = (0 - 5)^2 - 4 = 25 - 4 = 21
$$
- Y-intercept: $ (0, 21) $

#### X-intercepts:
Set $ y = 0 $:
$$
0 = (x - 5)^2 - 4 \\
(x - 5)^2 = 4 \\
x - 5 = \pm 2 \\
x = 5 \pm 2 \Rightarrow x = 3 \text{ or } 7
$$
- X-intercepts: $ (3, 0) $, $ (7, 0) $

#### Graph:
- Vertex at $ (5, -4) $
- Opens up
- Passes through $ (3,0), (7,0), (0,21) $
- Sketch: U-shaped with vertex at bottom

---

2) $ y = -(x + 3)^2 - 1 $

Rewrite as: $ y = -(x - (-3))^2 - 1 $

#### Vertex:
- $ h = -3 $, $ k = -1 $
- Vertex: $ (-3, -1) $

#### Axis of Symmetry:
- $ x = -3 $

#### Direction of Opening:
- $ a = -1 < 0 $ → Opens downward

#### Min/Max Value:
- Maximum value: $ y = -1 $

#### Y-intercept:
Set $ x = 0 $:
$$
y = -(0 + 3)^2 - 1 = -9 - 1 = -10
$$
- Y-intercept: $ (0, -10) $

#### X-intercepts:
Set $ y = 0 $:
$$
0 = -(x + 3)^2 - 1 \\
(x + 3)^2 = -1
$$
No real solution → No x-intercepts

#### Graph:
- Vertex at $ (-3, -1) $
- Opens downward
- No x-intercepts
- Passes through $ (0, -10) $
- Inverted U-shape

---

3) $ y = -(x - 1)^2 + 1 $

#### Vertex:
- $ h = 1 $, $ k = 1 $
- Vertex: $ (1, 1) $

#### Axis of Symmetry:
- $ x = 1 $

#### Direction of Opening:
- $ a = -1 < 0 $ → Opens downward

#### Min/Max Value:
- Maximum: $ y = 1 $

#### Y-intercept:
Set $ x = 0 $:
$$
y = -(0 - 1)^2 + 1 = -1 + 1 = 0
$$
- Y-intercept: $ (0, 0) $

#### X-intercepts:
Set $ y = 0 $:
$$
0 = -(x - 1)^2 + 1 \\
(x - 1)^2 = 1 \\
x - 1 = \pm 1 \\
x = 0 \text{ or } 2
$$
- X-intercepts: $ (0, 0) $, $ (2, 0) $

#### Graph:
- Vertex at $ (1,1) $
- Opens down
- Passes through $ (0,0), (2,0) $
- Symmetric about $ x = 1 $

---

4) $ y = (x + 1)^2 + 3 $

Rewrite: $ y = (x - (-1))^2 + 3 $

#### Vertex:
- $ h = -1 $, $ k = 3 $
- Vertex: $ (-1, 3) $

#### Axis of Symmetry:
- $ x = -1 $

#### Direction of Opening:
- $ a = 1 > 0 $ → Opens upward

#### Min/Max Value:
- Minimum: $ y = 3 $

#### Y-intercept:
Set $ x = 0 $:
$$
y = (0 + 1)^2 + 3 = 1 + 3 = 4
$$
- Y-intercept: $ (0, 4) $

#### X-intercepts:
Set $ y = 0 $:
$$
0 = (x + 1)^2 + 3 \\
(x + 1)^2 = -3
$$
No real solution → No x-intercepts

#### Graph:
- Vertex at $ (-1, 3) $
- Opens upward
- No x-intercepts
- Passes through $ (0, 4) $
- U-shaped above x-axis

---

5) $ y = -(x + 2)^2 + 1 $

Rewrite: $ y = -(x - (-2))^2 + 1 $

#### Vertex:
- $ h = -2 $, $ k = 1 $
- Vertex: $ (-2, 1) $

#### Axis of Symmetry:
- $ x = -2 $

#### Direction of Opening:
- $ a = -1 < 0 $ → Opens downward

#### Min/Max Value:
- Maximum: $ y = 1 $

#### Y-intercept:
Set $ x = 0 $:
$$
y = -(0 + 2)^2 + 1 = -4 + 1 = -3
$$
- Y-intercept: $ (0, -3) $

#### X-intercepts:
Set $ y = 0 $:
$$
0 = -(x + 2)^2 + 1 \\
(x + 2)^2 = 1 \\
x + 2 = \pm 1 \\
x = -1 \text{ or } -3
$$
- X-intercepts: $ (-3, 0) $, $ (-1, 0) $

#### Graph:
- Vertex at $ (-2, 1) $
- Opens down
- Passes through $ (-3,0), (-1,0), (0,-3) $
- Symmetric about $ x = -2 $

---

6) $ y = -(x - 5)^2 $

This is $ y = -(x - 5)^2 + 0 $

#### Vertex:
- $ h = 5 $, $ k = 0 $
- Vertex: $ (5, 0) $

#### Axis of Symmetry:
- $ x = 5 $

#### Direction of Opening:
- $ a = -1 < 0 $ → Opens downward

#### Min/Max Value:
- Maximum: $ y = 0 $

#### Y-intercept:
Set $ x = 0 $:
$$
y = -(0 - 5)^2 = -25
$$
- Y-intercept: $ (0, -25) $

#### X-intercepts:
Set $ y = 0 $:
$$
0 = -(x - 5)^2 \\
(x - 5)^2 = 0 \\
x = 5
$$
- X-intercept: $ (5, 0) $ → touches x-axis at vertex

#### Graph:
- Vertex at $ (5, 0) $
- Opens downward
- Touches x-axis at $ (5, 0) $
- Passes through $ (0, -25) $
- Only one x-intercept (tangent to x-axis)

---

Summary Table

| Problem | Vertex | Axis of Symmetry | Direction | Min/Max | Y-int | X-int |
|--------|--------|------------------|-----------|---------|-------|--------|
| 1 | (5, -4) | x = 5 | Up | Min: -4 | (0, 21) | (3,0), (7,0) |
| 2 | (-3, -1) | x = -3 | Down | Max: -1 | (0, -10) | None |
| 3 | (1, 1) | x = 1 | Down | Max: 1 | (0, 0) | (0,0), (2,0) |
| 4 | (-1, 3) | x = -1 | Up | Min: 3 | (0, 4) | None |
| 5 | (-2, 1) | x = -2 | Down | Max: 1 | (0, -3) | (-3,0), (-1,0) |
| 6 | (5, 0) | x = 5 | Down | Max: 0 | (0, -25) | (5,0) |

---

Final Step: Sketching the Graphs

Since you're supposed to sketch them on the grid, here’s how to do it quickly:

- Plot the vertex
- Draw the axis of symmetry (vertical dashed line)
- Use the direction of opening
- Plot the y-intercept
- Plot x-intercepts if any
- Use symmetry to find more points

For example:
- For #1: Plot (5, -4), then go up 1 unit to (5, -3), left/right 1 unit to (4, -3), (6, -3), etc.
- Connect smoothly into a parabola.

You can now complete the graphs based on this analysis.

Let me know if you'd like me to generate the sketches digitally!