To draw the graph of the given quadratic function written in standard form, we must start by identifying the values of a, b, and c.
y= 12x^2 ⇔ y=12x^2+ x+
We can see that a=12, b= , and c= . Now, we will follow four steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=0.
Calculating the Vertex
To calculate the vertex, we 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 ) )
Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=0. Thus, the x-coordinate of the vertex is also 0. To find the y-coordinate, we need to substitute 0 for x in the given equation.
We found the y-coordinate, and now we know that the vertex is (0,0).
Identifying the y-intercept and Other Points
The y-intercept of the graph of a quadratic function written in standard form is given by the value of c. Thus, the point where our graph intercepts the y-axis is (0, ), which is the same as the vertex. We are going to look for two more points that belong to the graph by substituting - 2 and 2 into the function rule.
x
1/2x^2
y=1/2x^2
- 2
1/2( - 2)^2
2
2
1/2( 2)^2
2
We found that (- 2,2) and (2,2) lie on the graph. Note that these two points are the reflection of each other across the axis of symmetry!
Connecting the Points
We can now draw the graph of the function. Since a=12, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.
