Finding the Vertex, Axis of Symmetry, and Intercepts of the Quadratic Equation
The quadratic equation we have is y = x² + 6x + 5. To analyze this quadratic function, we will determine its vertex, axis of symmetry, and intercepts systematically.
1. Finding the Vertex
The vertex of a parabola given in standard form y = ax² + bx + c can be found using the formula:
x = -b / 2a
For our equation:
- a = 1
- b = 6
- c = 5
Now, substituting the values of a and b into the formula:
x = -6 / (2 * 1) = -6 / 2 = -3
Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting x = -3 back into the original equation:
y = (-3)² + 6(-3) + 5
y = 9 – 18 + 5 = -4
Thus, the vertex is at the point (-3, -4).
2. Axis of Symmetry
The axis of symmetry for a quadratic function is a vertical line that passes through the vertex. The equation of the axis of symmetry can be expressed as:
x = -3
This means that the parabola is symmetrical about the line x = -3.
3. Finding the Intercepts
Next, let’s find the x-intercepts and y-intercept.
a. Finding the x-intercepts
The x-intercepts occur when y = 0.
To find the x-intercepts, we set the equation to zero:
0 = x² + 6x + 5
Next, we can factor the quadratic equation:
0 = (x + 1)(x + 5)
Setting each factor equal to zero gives:
- x + 1 = 0 ⇒ x = -1
- x + 5 = 0 ⇒ x = -5
Thus, the x-intercepts are at the points (-1, 0) and (-5, 0).
b. Finding the y-intercept
The y-intercept occurs when x = 0.
Substituting x = 0 into the original equation:
y = 0² + 6(0) + 5 = 5
Therefore, the y-intercept is at the point (0, 5).
Summary
- Vertex: (-3, -4)
- Axis of Symmetry: x = -3
- x-intercepts: (-1, 0), (-5, 0)
- y-intercept: (0, 5)
This analysis provides a clear understanding of the function’s key characteristics!