(x-1)(x-5) Graph

4 min read Jun 17, 2024

Graphing the Function (x-1)(x-5)

The function (x-1)(x-5) is a quadratic function, which means its graph is a parabola. Let's explore how to graph this function:

1. Finding the x-intercepts

The x-intercepts are the points where the graph crosses the x-axis. These occur when y = 0. Therefore, to find the x-intercepts, we set the function equal to zero and solve for x:

(x-1)(x-5) = 0

This equation is true when either (x-1) = 0 or (x-5) = 0. Solving for x, we find:

x = 1
x = 5

This means the graph intersects the x-axis at the points (1, 0) and (5, 0).

2. Finding the y-intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when x = 0. Substituting x = 0 into the function, we get:

(0-1)(0-5) = 5

Therefore, the y-intercept is at the point (0, 5).

3. Determining the Vertex

The vertex is the point where the parabola reaches its maximum or minimum value. We can find the x-coordinate of the vertex using the formula:

x = -b / 2a

where a and b are the coefficients of the quadratic function when it's written in standard form (ax² + bx + c).

Expanding the given function, we get:

(x-1)(x-5) = x² - 6x + 5

Therefore, a = 1 and b = -6. Plugging these values into the formula, we find:

x = -(-6) / (2 * 1) = 3

To find the y-coordinate of the vertex, we substitute x = 3 back into the original function:

(3 - 1)(3 - 5) = -4

Therefore, the vertex of the parabola is at the point (3, -4).

4. Plotting the Points and Drawing the Graph

Now that we know the x-intercepts, y-intercept, and vertex, we can plot these points on a graph. Since the coefficient of the x² term is positive, the parabola opens upwards. Connect the points with a smooth curve, ensuring the curve passes through all the plotted points and maintains the upward shape.

5. Additional Features

Axis of Symmetry: The vertical line that passes through the vertex is the axis of symmetry. In this case, the axis of symmetry is the line x = 3.
Domain and Range: The domain of the function is all real numbers, as the parabola extends infinitely in both directions. The range, however, is restricted since the parabola has a minimum point. The range is all real numbers greater than or equal to -4.

By following these steps, you can accurately graph the function (x-1)(x-5).