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 the y- and its symmetric point 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=- 3x^2-2x+1
⇕
y= - 3x^2+( - 2)x+ 1
We have identified that a= - 3, b= - 2, and c= 1.
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= - 3 and b= - 2 into this equation.
x=- b/2a
x=- - 2/2( - 3)
x=-1/3
The axis of symmetry is the line x=- 13.
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. Note that the vertex lies on the axis of symmetry x=- 13.
Vertex: ( - b/2a, f(- b/2a ) )
When determining the axis of symmetry, we found that - b2a=- 13. Therefore, the x-coordinate of the vertex is - 13 and the y-coordinate is f(- 13). To find this value, we will substitute our x-coordinate for x in the given equation.
y=- 3x^2-2x+1
y=- 3( -1/3) ^2-2( -1/3) +1
y=- 3(-1/3) ^2-2(-1/3) +1
y=- 3(1/9)-2(-1/3) +1
y=- 3(1/9)+2*1/3+1
y=-3/9+2/3+1
y=-1/3+2/3+1
y=1/3+1
y=1 13
The vertex of the parabola is (- 13,1 13).
y-intercept and Symmetric Point
For a function written in standard form, the y-intercept is (0, c). Since, in our equation, we have that c= 1, the y-intercept is (0, 1). Let's plot this point and its reflection across the axis of symmetry.
Graph
Since a= - 3, which is less than zero, we know that our parabola opens downwards. Let's draw a smooth curve connecting the three points we have. You should not use a straight edge for this!
One common mistake when identifying the key features of a parabola algebraically is forgetting to include the negatives in the values of these constants. The standard form is addition only, so any subtraction must be treated as negative values of a, b, or c. Let's look at an example.
y=3x^2-4x-2 ⇕ y=3x^2 + (-4x) + (-2)
In this case, the values of a, b, and c are 3, -4, and -2. They are NOT 3, 4, and 2.
a=3, b=4, c=2 *
a=3, b=-4, c=-2 ✓
