We will continue with Unit 5 Graphing Quadratics. We will also start with unit 6 solving leaner equations.
Unit 5 Solving leaner equations when you graph on the X and Y axis
When graphed, quadratic equations of the form ax2 + bx + c or a(x - h)2 + k give a smooth U-shaped or a reverse U-shaped curve called a parabola. Graphing a quadratic equation is a matter of finding its vertex, direction, and, often, its x and y intercepts. In the cases of relatively simple quadratic equations, it may also be enough to plug in a range of x values and plot a curve based on the resulting points. See Step 1 below to get started.
Step 1
Determine which form of quadratic equation you have. The quadratic equation can be written in three different forms: the standard form, vertex form, and the quadratic form. You can use either form to graph a quadratic equation; the process for graphing each is slightly different. If you're doing a homework problem, you'll usually receive the problem in one of these two forms - in other words, you won't be able to choose, so it's best to understand both. The two forms of quadratic equation are:- Standard form. In this form, the quadratic equation is written as: f(x) = ax2 + bx + c where a, b, and c are real numbers and a is not equal to zero.
- For example, two standard form quadratic equations are f(x) = x2 + 2x + 1 and f(x) = 9x2 + 10x -8.
- Vertex form. In this form, the quadratic equation is written as: f(x) = a(x - h)2 + k where a, h, and k are real numbers and a does not equal zero. Vertex form is so named because h and k directly give you the vertex (central point) of your parabola at the point (h,k).
Step 2
Define your variables. To be able to solve a quadratic problem, the variables a, b, and c (or a, h, and k) usually need to be defined.
Step 3
Calculate h. In vertex form equations, your value for h is already given, but in standard form equations, it must be calculated. Remember that, for standard form equations, h = -b/2a.
Step 4
Calculate k. As with h, k is already known in vertex form equations. For standard form equations, remember that k = f(h). In other words, you can find k by replacing every instance of x in your equation with the value you just found for h.
Step 5
Plot your vertex. The vertex of your parabola will be the point (h, k) - h specifies the x coordinate, while k specifies the y coordinate. The vertex is the central point in your parabola - either the very bottom of a "U" or the very top of an upside-down "U."
Step 6
Draw the parabola's axis (optional). A parabola's axis of symmetry is the line running through its middle which divides it perfectly in half. Across this axis, the left side of the parabola will mirror the right side.
- For example, two standard form quadratic equations are f(x) = x2 + 2x + 1 and f(x) = 9x2 + 10x -8.