Graphing Quadratic Functions
Graphing quadratic functions is a process of plotting quadratic functions in a coordinate plane. The graph of a quadratic function is a parabola and tells the nature of the quadratic function. Graphing quadratic equations gives parabolas that are U-shaped, and wide or narrow depending upon the coefficients of the function.
In this article, we will explore the graphs of quadratic functions. We will learn quadratic graph, vertex form of quadratic function, graphing quadratic functions in vertex form, standard form of a quadratic function, graphing quadratic functions in standard form, quadratic function graph, and other interesting facts around the topic.
| 1. | What is Graphing Quadratic Functions? |
| 2. | Graphing Quadratic Functions in Vertex Form |
| 3. | Graphing Quadratic Functions in Standard Form |
| 4. | FAQs on Graphing Quadratic Functions |
What is Graphing Quadratic Functions?
Graphing quadratic functions is a technique to study the nature of the quadratic functions graphically. The shape of the parabola (graph of a quadratic function) is determined by the coefficient 'a' of the quadratic function f(x) = ax2 + bx + c, where a, b, c are real numbers and a ≠ 0. Now, in terms of graphing quadratic functions, we will understand a step-by-step procedure to plot the graph of any quadratic function.
The vertex form of a quadratic function is f(x) = a(x - h)2 + k, where (h, k) is the vertex of the parabola. The coefficient a determines whether the graph of a quadratic function will open upwards or downwards.
Graphing Quadratic Functions in Vertex Form
We will study a step-by-step procedure to plot the graph of any quadratic function. Consider the general quadratic function f(x) = ax2 + bx + c. First, we rearrange it (by the method of completion of squares) to the following form: f(x) = a(x + b/2a)2 - D/4a. The term D is the discriminant, given by D = b2 - 4ac. Here, the vertex of the parabola is (h, k) = (-b/2a, -D/4a). Now, to plot the graph of f(x), we start by taking the graph of x2, and applying a series of transformations to it:
- Step 1: x2 to ax2: This will imply a vertical scaling of the original parabola. If a is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of a.
- Step 2: ax2 to a(x + b/2a)2: This is a horizontal shift of magnitude |b/2a| units. The direction of the shift will be decided by the sign of b/2a. The new vertex of the parabola will be at (-b/2a,0). The following figure shows an example shift:
- Step 3: a(x + b/2a)2 to a(x + b/2a)2 - D/4a: This transformation is a vertical shift of magnitude |D/4a| units. The direction of the shift will be decided by the sign of D/4a. The final vertex of the parabola will be at (-b/2a, -D/4a). The following figure shows an example shift:
The graph of quadratic functions can also be obtained using the graphing quadratic functions calculator.
Graphing Quadratic Functions in Standard Form
The general equation of a quadratic function is f(x) = ax2 + bx + c. Now, for graphing quadratic functions using the standard form of the function, we can either convert the general form to the vertex form and then plot the graph of the quadratic function, or determine the axis of symmetry and y-intercept of the graph and plot it.
For example, we have a quadratic function f(x) = 2x2 + 4x + 4. The coefficient a = 2 > 0, implies the graph of the quadratic function will open upwards. The coefficient a also controls the speed of increase (or decrease) of the graph of the quadratic function from the vertex. A larger and positive 'a' makes the function increase faster and the graph appear thinner.
Next, for graphing quadratic function f(x) = 2x2 + 4x + 4, we determine the axis of symmetry of the parabola which is given by, x = -b/2a = -4/(2.2) = -1. Hence, x = -1 is the axis of symmetry for the graph of f(x) = 2x2 + 4x + 4 and the vertex of the graph also has x-coordinate equal to -1.
Now, we will determine the y-intercept of the parabola which is given by (0, c) = (0, 4). Using all this information, we can plot the graph of the quadratic function f(x) = 2x2 + 4x + 4.
Important Notes on Graphing Quadratic Functions
- The graph of the quadratic function is in the form of a parabola.
- The coefficient a in f(x) = a(x - h)2 + k determines whether the graph of a quadratic function will open upwards or downwards.
- Graphing Quadratic Functions can be done using both general form and vertex form.
Related Topics on Graphing Quadratic Functions
Graphing Quadratic Functions Examples
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Example 1: Plot the graph of quadratic function f(x) = 1- 2x - 3x2 using graphing qudratic functions in vertex form.
Solution:
f(x) = 1 - 2x - 3x2 ⇒ a = - 3, b = - 2, c = 1, D = b2 - 4ac = 16. The coordinates of the vertex are: V = (-b/2a, -D/4a) = (-1/3, 4/3) = (-0.333, 1.333)
Note that the parabola will open downward (since a is negative), but the vertex has a positive y-coordinate. This means that the parabola crosses the x-axis. In other words, the quadratic function as real zeroes. Let us calculate these zeroes:
x = [-(-2) ± √(16)]/[2.(-3)] = - 1, -1/3. The parabola will cross the x-axis at these x values. Finally, using all this information, we plot the quadratic graph.
FAQs on Graphing Quadratic Functions
What is Graphing Quadratic Functions?
Graphing quadratic functions is a process of plotting quadratic functions in a coordinate plane. Graphing quadratic functions is a technique to study the nature of the quadratic functions graphically.
How to Do Graphing Quadratic Functions?
- Step 1: x2 to ax2: This will imply a vertical scaling of the original parabola. If a is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of a.
- Step 2: ax2 to a(x + b/2a)2: This is a horizontal shift of magnitude |b/2a| units. The direction of the shift will be decided by the sign of b/2a. The new vertex of the parabola will be at (-b/2a,0).
- Step 3: a(x + b/2a)2 to a(x + b/2a)2 - D/4a: This transformation is a vertical shift of magnitude |D/4a| units. The direction of the shift will be decided by the sign of D/4a. The final vertex of the parabola will be at (-b/2a, -D/4a).
How to Find the Vertex when Graphing Quadratic Functions?
To determine the vertex of the parabola when graphing quadratic equations, we determine the x-coordinate of the vertex using the formula x = -b/2a. Then, substitute this value of x in the quadratic function f(x) = ax2 + bx + c to determine the y-coordinate of the vertex.
How to Do Graphing Quadratic Functions In Standard Form?
For graphing quadratic functions using the standard form of the function, we can either convert the general form to the vertex form and then plot the graph of the quadratic function, or determine the axis of symmetry and y-intercept of the graph and plot it.
What is a Real-Life Example of Graphing Quadratic Functions?
When a ball is thrown in the air, its path is modeled by a quadratic function.
What is the Shape for Graphing Quadratic Functions?
The quadratic graphs are shaped as parabolas.
How to Solve Graphing Quadratic Functions?
The x-intercepts of the graphs give the solution of the quadratic functions.