Graph the following equations and explain why, if they are not graphs of functions of x
a) ƒ(x)=1-2x-x^2

Let's start by analyzing the given equation:

### a) \( f(x) = 1 - 2x - x^2 \)

#### Step 1: Identifying the Type of Equation
This is a quadratic equation of the form:
\[
f(x) = ax^2 + bx + c
\]
In this case:
- \( a = -1 \)
- \( b = -2 \)
- \( c = 1 \)

Quadratic equations are parabolas, and since \( a = -1 \) (which is negative), the parabola opens **downwards**.

#### Step 2: Graphing the Equation
1. **Vertex**: The vertex of a parabola can be found using the formula:
   \[
   x_{\text{vertex}} = -\frac{b}{2a} = -\frac{-2}{2(-1)} = 1
   \]
   Plugging \( x = 1 \) back into the equation to find \( f(1) \):
   \[
   f(1) = 1 - 2(1) - (1)^2 = 1 - 2 - 1 = -2
   \]
   So, the vertex is at \( (1, -2) \).

2. **Axis of Symmetry**: Since the parabola is symmetric around \( x = 1 \), the axis of symmetry is the vertical line \( x = 1 \).

3. **Y-Intercept**: The y-intercept occurs when \( x = 0 \):
   \[
   f(0) = 1 - 2(0) - (0)^2 = 1
   \]
   So, the y-intercept is at \( (0, 1) \).

4. **Additional Points**: Let's compute a couple of additional points to sketch the curve better:
   - For \( x = -1 \):
     \[
     f(-1) = 1 - 2(-1) - (-1)^2 = 1 + 2 - 1 = 2
     \]
     Point: \( (-1, 2) \)
   - For \( x = 2 \):
     \[
     f(2) = 1 - 2(2) - (2)^2 = 1 - 4 - 4 = -7
     \]
     Point: \( (2, -7) \)

#### Step 3: Is it a Function of \( x \)?
To determine whether this is a function of \( x \), we apply the **vertical line test**: If a vertical line intersects the graph at more than one point, then it is **not a function**.

Since this is a parabola, and for every value of \( x \) there is only **one** corresponding value of \( f(x) \), it **passes the vertical line test**. Therefore, \( f(x) = 1 - 2x - x^2 \) **is a function** of \( x \).

Would you like me to graph this function for you?

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### Related Questions:
1. How do you determine the direction in which a parabola opens?
2. What is the importance of the vertex in a quadratic function?
3. How do you find the axis of symmetry in any parabola?
4. How does the vertical line test help in determining if a relation is a function?
5. Can a quadratic equation ever fail the vertical line test?

**Tip**: The sign of \( a \) in a quadratic equation \( ax^2 + bx + c \) controls whether the parabola opens upwards (positive \( a \)) or downwards (negative \( a \)).

Graph the following equations and explain why, if they are not graphs of functions of x. a) ƒ(x)=1-2x-x^2

Learn how to graph the quadratic equation f(x) = 1 - 2x - x^2, find its vertex, axis of symmetry, and y-intercept. Understand why it passes the vertical line test, confirming it is a function. This step-by-step explanation is suitable for students in Grades 9-11.

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