Step 1: Understand the Equation
The given equation is in the form of y = a(x - r)(x - s), where a is the coefficient, and r and s are the roots of the equation.
Here, a = -\dfrac{5}{4}, and the roots are r = 1 and s = -3.
Understanding the equation in this form will help us visualize the graph.
Step 2: Determine the Vertex
The vertex of the parabola is the point where the graph changes direction.
For the equation y = a(x - r)(x - s), the vertex is located at the midpoint of the roots.
So, the x-coordinate of the vertex is x = \dfrac{r + s}{2} = \dfrac{1 + (-3)}{2} = -1.
Now that we have the x-coordinate, we can find the y-coordinate by substituting x into the equation.
y = -\dfrac{5}{4}(x - 1)(x + 3) becomes y = -\dfrac{5}{4}(-1 - 1)(-1 + 3) = -\dfrac{5}{4}(-2)(2) = 10.
Therefore, the vertex is located at (-1, 10).
Step 3: Identify the Roots and Intercepts
The roots of the equation are the x-intercepts of the graph.
From the equation, we can see that the roots are r = 1 and s = -3.
The x-intercepts are the points where the graph crosses the x-axis, and their coordinates are (1, 0) and (-3, 0).
Next, let's find the y-intercept by substituting x = 0 into the equation.
y = -\dfrac{5}{4}(0 - 1)(0 + 3) = -\dfrac{5}{4}(-1)(3) = \dfrac{15}{4}.
So, the y-intercept is located at (0, \dfrac{15}{4}).
Step 4: Plot the Graph
Now that we have the vertex, roots, and intercepts, we can start plotting the graph.
Begin by plotting the vertex at (-1, 10).
Next, plot the roots at (1, 0) and (-3, 0).
Then, plot the y-intercept at (0, \dfrac{15}{4}).
Finally, use a ruler or a straightedge to draw a smooth curve through the points, making sure to include at least two points on either side of the vertex.
Table: Comparing Different Quadratic Equations
| Equation | Vertex | Roots | Y-intercept |
|---|---|---|---|
| y = x^2 - 4x + 4 | (2, 0) | Root 1: 2, Root 2: 2 | Y-intercept: 0 |
| y = -2(x - 1)(x + 2) | (-1, 4) | Root 1: 1, Root 2: -2 | Y-intercept: -4 |
| y = \dfrac{1}{2}(x - 2)^2 + 3 | (2, 3) | Root: 2 | Y-intercept: 3 |
By comparing the equations and their corresponding graphs, we can see how different coefficients and roots affect the shape and position of the parabola.
Tips and Tricks
- When graphing a quadratic equation, it's essential to identify the vertex, roots, and intercepts.
- Use a ruler or a straightedge to draw a smooth curve through the points, making sure to include at least two points on either side of the vertex.
- Pay attention to the direction of the parabola, as it can change direction at the vertex.
- Use the table to compare different quadratic equations and their corresponding graphs.
- Practice graphing different quadratic equations to become more comfortable with the process.
Conclusion
Graphing a quadratic equation, such as y = -\dfrac{5}{4}(x - 1)(x + 3), requires a thorough understanding of the equation and its components.
By following the steps outlined in this guide, you should be able to graph the equation accurately and efficiently.
Remember to practice graphing different quadratic equations to become more confident and proficient in your ability to graph these equations.