We want to solve the given by graphing the related function. The solutions of these equations are the x-coordinates of the of the and the x-axis. Recall that a quadratic equation can have two, one, or no solutions.
Graphing the Related Function
We want to draw the graph of a written in .
y=ax^2+bx+c
To do so, we will follow five steps.
- Identify a, b, and c.
- Calculate and sketch the .
- Find and plot the .
- Find and plot two other symmetric points across the axis of symmetry.
- Draw a smooth curve through the three plotted points.
Let's do it!
Identify a, b, and c
We will start by identifying the values of a, b, and c.
y=0.25x^2
⇕
y= 0.25x^2+ x+
We have identified that a= 0.25, b= , and c= .
Axis of Symmetry
The axis of symmetry is the that divides the into two mirror images. Its equation follows a specific formula.
x=- b/2 a
Let's substitute our given values a= 0.25 and b= into this equation.
x=- b/2a
x=- /2( 0.25)
x=- 0/0.5
x=- 0
x = 0
The axis of symmetry is the line x=0, thus the graph is symmetric about the y-axis.
Vertex
To find the vertex of the parabola, we will need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b.
Vertex: ( - b/2a, f(- b/2a ) )
When determining the axis of symmetry, we found that - b2a=0. Therefore, the x-coordinate of the vertex is 0 and the y-coordinate is f(0). To find this value, substitute our x-coordinate for x in the given equation.
y=0.25x^2
y=0.25( 0)^2
y=0.25(0)
y=0
The vertex of the parabola is (0,0).
Symmetric Points Across the Axis of Symmetry
Since the graph is symmetric about the y-axis, we can find the axis of symmetry by plotting two points equal in distance from x=0. We are going to find such points by substituting - 4 and 4 into the function rule. The resulting (-4,f(- 4)) and (4,f(4)) are the coordinates of the points.
| x
|
0.25x^2
|
y=0.25x^2
|
| - 4
|
0.25( - 4)^2
|
4
|
| 4
|
0.25( 4)^2
|
4
|
We found that (- 4,4) and (4,4) lie on the graph. Let's now plot both points.
Graph
Since a=0.25, which is greater than zero, we can confirm that our parabola opens upwards.
Let's draw a smooth curve connecting the three points we have. You should not use a straight edge for this!
Determining the Real Solutions
Let's consider the graph.
This graph touches the x-axis at one point, so its related equation 0.25x^2=0 has one solution. We can see that the solution is x=0.
