Practice Worksheet Graphing Quadratic Functions In Intercept Form ... - Free Printable

To solve the problems involving quadratic functions in intercept form, we need to analyze each function step by step. The general intercept form of a quadratic function is:

\[
y = a(x - p)(x - q)
\]

where:
- \( p \) and \( q \) are the x-intercepts.
- The axis of symmetry is the vertical line \( x = \frac{p + q}{2} \).
- The vertex lies on the axis of symmetry.
- The y-intercept is found by setting \( x = 0 \).

Let's solve each problem systematically.

---

Problem 1: \( y = \frac{1}{2}(x + 4)(x - 2) \)

#### Step 1: Identify the x-intercepts
The x-intercepts occur where \( y = 0 \). Set \( y = 0 \):

\[
0 = \frac{1}{2}(x + 4)(x - 2)
\]

This gives us:

\[
x + 4 = 0 \quad \text{or} \quad x - 2 = 0
\]

\[
x = -4 \quad \text{or} \quad x = 2
\]

So, the x-intercepts are \( (-4, 0) \) and \( (2, 0) \).

#### Step 2: Find the axis of symmetry
The axis of symmetry is the midpoint of the x-intercepts:

\[
x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1
\]

So, the axis of symmetry is \( x = -1 \).

#### Step 3: Find the vertex
The vertex lies on the axis of symmetry \( x = -1 \). Substitute \( x = -1 \) into the equation to find the y-coordinate:

\[
y = \frac{1}{2}((-1) + 4)((-1) - 2)
\]

\[
y = \frac{1}{2}(3)(-3) = \frac{1}{2} \cdot (-9) = -\frac{9}{2}
\]

So, the vertex is \( \left( -1, -\frac{9}{2} \right) \).

#### Step 4: Find the y-intercept
The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = \frac{1}{2}(0 + 4)(0 - 2)
\]

\[
y = \frac{1}{2}(4)(-2) = \frac{1}{2} \cdot (-8) = -4
\]

So, the y-intercept is \( (0, -4) \).

#### Summary for Problem 1:
- x-intercepts: \( (-4, 0) \) and \( (2, 0) \)
- Axis of symmetry: \( x = -1 \)
- Vertex: \( \left( -1, -\frac{9}{2} \right) \)
- y-intercept: \( (0, -4) \)

---

Problem 2: \( y = 6(x + 2)(x - 2) \)

#### Step 1: Identify the x-intercepts
Set \( y = 0 \):

\[
0 = 6(x + 2)(x - 2)
\]

This gives us:

\[
x + 2 = 0 \quad \text{or} \quad x - 2 = 0
\]

\[
x = -2 \quad \text{or} \quad x = 2
\]

So, the x-intercepts are \( (-2, 0) \) and \( (2, 0) \).

#### Step 2: Find the axis of symmetry
The axis of symmetry is the midpoint of the x-intercepts:

\[
x = \frac{-2 + 2}{2} = \frac{0}{2} = 0
\]

So, the axis of symmetry is \( x = 0 \).

#### Step 3: Find the vertex
The vertex lies on the axis of symmetry \( x = 0 \). Substitute \( x = 0 \) into the equation to find the y-coordinate:

\[
y = 6(0 + 2)(0 - 2)
\]

\[
y = 6(2)(-2) = 6 \cdot (-4) = -24
\]

So, the vertex is \( (0, -24) \).

#### Step 4: Find the y-intercept
The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = 6(0 + 2)(0 - 2)
\]

\[
y = 6(2)(-2) = 6 \cdot (-4) = -24
\]

So, the y-intercept is \( (0, -24) \).

#### Summary for Problem 2:
- x-intercepts: \( (-2, 0) \) and \( (2, 0) \)
- Axis of symmetry: \( x = 0 \)
- Vertex: \( (0, -24) \)
- y-intercept: \( (0, -24) \)

---

Problem 3: \( y = -\frac{1}{2}(x + 1)(x - 5) \)

#### Step 1: Identify the x-intercepts
Set \( y = 0 \):

\[
0 = -\frac{1}{2}(x + 1)(x - 5)
\]

This gives us:

\[
x + 1 = 0 \quad \text{or} \quad x - 5 = 0
\]

\[
x = -1 \quad \text{or} \quad x = 5
\]

So, the x-intercepts are \( (-1, 0) \) and \( (5, 0) \).

#### Step 2: Find the axis of symmetry
The axis of symmetry is the midpoint of the x-intercepts:

\[
x = \frac{-1 + 5}{2} = \frac{4}{2} = 2
\]

So, the axis of symmetry is \( x = 2 \).

#### Step 3: Find the vertex
The vertex lies on the axis of symmetry \( x = 2 \). Substitute \( x = 2 \) into the equation to find the y-coordinate:

\[
y = -\frac{1}{2}(2 + 1)(2 - 5)
\]

\[
y = -\frac{1}{2}(3)(-3) = -\frac{1}{2} \cdot (-9) = \frac{9}{2}
\]

So, the vertex is \( \left( 2, \frac{9}{2} \right) \).

#### Step 4: Find the y-intercept
The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = -\frac{1}{2}(0 + 1)(0 - 5)
\]

\[
y = -\frac{1}{2}(1)(-5) = -\frac{1}{2} \cdot (-5) = \frac{5}{2}
\]

So, the y-intercept is \( \left( 0, \frac{5}{2} \right) \).

#### Summary for Problem 3:
- x-intercepts: \( (-1, 0) \) and \( (5, 0) \)
- Axis of symmetry: \( x = 2 \)
- Vertex: \( \left( 2, \frac{9}{2} \right) \)
- y-intercept: \( \left( 0, \frac{5}{2} \right) \)

---

Problem 4: \( y = 4(x + 2)(x + 1) \)

#### Step 1: Identify the x-intercepts
Set \( y = 0 \):

\[
0 = 4(x + 2)(x + 1)
\]

This gives us:

\[
x + 2 = 0 \quad \text{or} \quad x + 1 = 0
\]

\[
x = -2 \quad \text{or} \quad x = -1
\]

So, the x-intercepts are \( (-2, 0) \) and \( (-1, 0) \).

#### Step 2: Find the axis of symmetry
The axis of symmetry is the midpoint of the x-intercepts:

\[
x = \frac{-2 + (-1)}{2} = \frac{-3}{2}
\]

So, the axis of symmetry is \( x = -\frac{3}{2} \).

#### Step 3: Find the vertex
The vertex lies on the axis of symmetry \( x = -\frac{3}{2} \). Substitute \( x = -\frac{3}{2} \) into the equation to find the y-coordinate:

\[
y = 4\left(-\frac{3}{2} + 2\right)\left(-\frac{3}{2} + 1\right)
\]

\[
y = 4\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right) = 4 \cdot \frac{1}{2} \cdot \left(-\frac{1}{2}\right) = 4 \cdot \left(-\frac{1}{4}\right) = -1
\]

So, the vertex is \( \left( -\frac{3}{2}, -1 \right) \).

#### Step 4: Find the y-intercept
The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = 4(0 + 2)(0 + 1)
\]

\[
y = 4(2)(1) = 4 \cdot 2 = 8
\]

So, the y-intercept is \( (0, 8) \).

#### Summary for Problem 4:
- x-intercepts: \( (-2, 0) \) and \( (-1, 0) \)
- Axis of symmetry: \( x = -\frac{3}{2} \)
- Vertex: \( \left( -\frac{3}{2}, -1 \right) \)
- y-intercept: \( (0, 8) \)

---

Final Answers

1. \( y = \frac{1}{2}(x + 4)(x - 2) \)
- x-intercepts: \( (-4, 0) \) and \( (2, 0) \)
- Axis of symmetry: \( x = -1 \)
- Vertex: \( \left( -1, -\frac{9}{2} \right) \)
- y-intercept: \( (0, -4) \)

2. \( y = 6(x + 2)(x - 2) \)
- x-intercepts: \( (-2, 0) \) and \( (2, 0) \)
- Axis of symmetry: \( x = 0 \)
- Vertex: \( (0, -24) \)
- y-intercept: \( (0, -24) \)

3. \( y = -\frac{1}{2}(x + 1)(x - 5) \)
- x-intercepts: \( (-1, 0) \) and \( (5, 0) \)
- Axis of symmetry: \( x = 2 \)
- Vertex: \( \left( 2, \frac{9}{2} \right) \)
- y-intercept: \( \left( 0, \frac{5}{2} \right) \)

4. \( y = 4(x + 2)(x + 1) \)
- x-intercepts: \( (-2, 0) \) and \( (-1, 0) \)
- Axis of symmetry: \( x = -\frac{3}{2} \)
- Vertex: \( \left( -\frac{3}{2}, -1 \right) \)
- y-intercept: \( (0, 8) \)

\boxed{
\begin{array}{l}
\text{1. x-intercepts: } (-4, 0), (2, 0); \text{ Axis of symmetry: } x = -1; \text{ Vertex: } \left( -1, -\frac{9}{2} \right); \text{ y-intercept: } (0, -4) \\
\text{2. x-intercepts: } (-2, 0), (2, 0); \text{ Axis of symmetry: } x = 0; \text{ Vertex: } (0, -24); \text{ y-intercept: } (0, -24) \\
\text{3. x-intercepts: } (-1, 0), (5, 0); \text{ Axis of symmetry: } x = 2; \text{ Vertex: } \left( 2, \frac{9}{2} \right); \text{ y-intercept: } \left( 0, \frac{5}{2} \right) \\
\text{4. x-intercepts: } (-2, 0), (-1, 0); \text{ Axis of symmetry: } x = -\frac{3}{2}; \text{ Vertex: } \left( -\frac{3}{2}, -1 \right); \text{ y-intercept: } (0, 8)
\end{array}
}