Drawing the Graph of y = x² – x
To draw the graph of the equation y = x² – x, we can start by determining some key features of the parabola defined by this quadratic function.
1. Identify the Vertex
The vertex form of a quadratic equation can be useful, but we can find the vertex directly from the standard form:
- Using the formula for the vertex, x = -b/(2a), where y = ax² + bx + c.
- For our equation, a = 1, b = -1, and c = 0.
- Thus, x = -(-1)/(2 * 1) = 1/2.
Substituting x = 1/2 back into the equation to find y gives:
- y = (1/2)² – (1/2) = 1/4 – 1/2 = -1/4.
So, the vertex of the parabola is at (1/2, -1/4).
2. Determine the Roots
Next, we find the points where the graph intersects the x-axis by setting y = 0:
- Setting the equation to zero: x² – x = 0.
- This can be factored as: x(x – 1) = 0.
- Thus, x = 0 and x = 1 are the roots.
3. Additional Points
To make the graph more accurate, let’s evaluate a few more points:
- x = -1: y = (-1)² – (-1) = 1 + 1 = 2
- x = 2: y = 2² – 2 = 4 – 2 = 2
4. Sketching the Graph
With this information:
- Plot the vertex at (1/2, -1/4).
- Mark the roots at (0, 0) and (1, 0).
- Plot additional points like (-1, 2) and (2, 2).
Now, you can draw a smooth curve through these points, forming a parabola opening upwards.
Solving the Equation x² – 1 = 0
Now let’s solve the equation x² – 1 = 0.
1. Factor the Equation
The left side can be factored:
- x² – 1 = (x – 1)(x + 1) = 0.
2. Find the solutions
Setting each factor equal to zero gives:
- x – 1 = 0 ⇒ x = 1
- x + 1 = 0 ⇒ x = -1
Thus, the solutions to the equation x² – 1 = 0 are x = 1 and x = -1.
Conclusion
In summary, the graph of y = x² – x is a parabola with a vertex at (1/2, -1/4) and intersects the x-axis at (0, 0) and (1, 0). The solutions to the equation x² – 1 = 0 are x = 1 and x = -1.