There are three forms of a quadratic function: standard, vertex, and intercept. Which form is best for each activity and explain why you think so.
14. Factor the equation
15. Graph the parabola
16. Identify the different properties of the graph
17. solve the equation using the quadratic formula
To determine which form of a quadratic function is best for each activity, we should consider the advantages and disadvantages of each form.
1. Standard form (ax^2 + bx + c):
- Factor the equation: The standard form can be factored easily if the coefficients are small and factorable. However, if the coefficients are large or prime, factoring might be more complicated.
2. Vertex form (a(x-h)^2 + k):
- Graph the parabola: The vertex form provides immediate information about the vertex of the parabola (the point (h, k)). This can make graphing the parabola much easier as you can easily identify the vertex without having to perform additional calculations.
3. Intercept form (a(x-r1)(x-r2)):
- Identify the different properties of the graph: The intercept form allows you to immediately determine the x-intercepts (roots) of the quadratic equation. This can help in identifying the number of real solutions, as well as the nature of those solutions (real or complex).
4. Solve the equation using the quadratic formula: The quadratic formula is in standard form (ax^2 + bx + c = 0), so using the quadratic formula is generally easier when the equation is already in standard form. However, it is always possible to convert the quadratic equation from vertex or intercept form to standard form before applying the quadratic formula.
In summary:
- To factor the equation (activity 14), it is best to have the quadratic equation in standard form.
- To graph the parabola (activity 15), it is best to use the vertex form as it gives immediate access to the coordinates of the vertex.
- To identify the different properties of the graph (activity 16), it is best to use the intercept form as it immediately gives access to the x-intercepts.
- To solve the equation using the quadratic formula (activity 17), it is usually best to use the standard form. However, you can convert equations from other forms to standard form if necessary.