Question
Explain in words and algebra how zeros the AOS and the vertex can be obtained: y=x^2+2x-8
Answers
GPT-4o mini
To find the zeros, axis of symmetry (AOS), and the vertex of the quadratic function \( y = x^2 + 2x - 8 \), we can start by analyzing the quadratic equation and using algebraic methods.
### Step 1: Finding the Zeros
The zeros of the quadratic function are the values of \( x \) where \( y = 0 \). We can solve for the zeros by setting the equation equal to zero:
\[
x^2 + 2x - 8 = 0
\]
To solve this quadratic equation, we can factor it if possible. We look for two numbers that multiply to \(-8\) (the constant term) and add to \(2\) (the coefficient of \(x\)).
The numbers \(4\) and \(-2\) fit these criteria:
\[
(x + 4)(x - 2) = 0
\]
Setting each factor equal to zero gives us:
1. \(x + 4 = 0 \Rightarrow x = -4\)
2. \(x - 2 = 0 \Rightarrow x = 2\)
Thus, the zeros of the function are:
\[
x = -4 \quad \text{and} \quad x = 2
\]
### Step 2: Finding the Axis of Symmetry (AOS)
The axis of symmetry for a quadratic function in the form \(y = ax^2 + bx + c\) is given by the formula:
\[
x = -\frac{b}{2a}
\]
In our case, \(a = 1\) and \(b = 2\). Plugging these values into the formula gives:
\[
x = -\frac{2}{2 \cdot 1} = -\frac{2}{2} = -1
\]
So, the axis of symmetry is:
\[
x = -1
\]
### Step 3: Finding the Vertex
The vertex of a parabola represented by the quadratic function can be found using the axis of symmetry. The \(x\)-coordinate of the vertex is the same as the AOS:
\[
x = -1
\]
Now, to find the \(y\)-coordinate of the vertex, we substitute \(x = -1\) back into the original equation:
\[
y = (-1)^2 + 2(-1) - 8
\]
\[
y = 1 - 2 - 8
\]
\[
y = -9
\]
Thus, the vertex of the parabola is:
\[
(-1, -9)
\]
### Summary
- **Zeros**: \(x = -4\) and \(x = 2\)
- **Axis of Symmetry**: \(x = -1\)
- **Vertex**: \((-1, -9)\)
In conclusion, we can derive the zeros, axis of symmetry, and vertex using algebraic methods of solving the quadratic equation and using the characteristics of quadratic functions.
### Step 1: Finding the Zeros
The zeros of the quadratic function are the values of \( x \) where \( y = 0 \). We can solve for the zeros by setting the equation equal to zero:
\[
x^2 + 2x - 8 = 0
\]
To solve this quadratic equation, we can factor it if possible. We look for two numbers that multiply to \(-8\) (the constant term) and add to \(2\) (the coefficient of \(x\)).
The numbers \(4\) and \(-2\) fit these criteria:
\[
(x + 4)(x - 2) = 0
\]
Setting each factor equal to zero gives us:
1. \(x + 4 = 0 \Rightarrow x = -4\)
2. \(x - 2 = 0 \Rightarrow x = 2\)
Thus, the zeros of the function are:
\[
x = -4 \quad \text{and} \quad x = 2
\]
### Step 2: Finding the Axis of Symmetry (AOS)
The axis of symmetry for a quadratic function in the form \(y = ax^2 + bx + c\) is given by the formula:
\[
x = -\frac{b}{2a}
\]
In our case, \(a = 1\) and \(b = 2\). Plugging these values into the formula gives:
\[
x = -\frac{2}{2 \cdot 1} = -\frac{2}{2} = -1
\]
So, the axis of symmetry is:
\[
x = -1
\]
### Step 3: Finding the Vertex
The vertex of a parabola represented by the quadratic function can be found using the axis of symmetry. The \(x\)-coordinate of the vertex is the same as the AOS:
\[
x = -1
\]
Now, to find the \(y\)-coordinate of the vertex, we substitute \(x = -1\) back into the original equation:
\[
y = (-1)^2 + 2(-1) - 8
\]
\[
y = 1 - 2 - 8
\]
\[
y = -9
\]
Thus, the vertex of the parabola is:
\[
(-1, -9)
\]
### Summary
- **Zeros**: \(x = -4\) and \(x = 2\)
- **Axis of Symmetry**: \(x = -1\)
- **Vertex**: \((-1, -9)\)
In conclusion, we can derive the zeros, axis of symmetry, and vertex using algebraic methods of solving the quadratic equation and using the characteristics of quadratic functions.