To draw the graph of the given , written in , we must start by identifying the values of
a, b, and
c.
f(x)=-x2−6x+3⇔f(x)=-1x2+(-6)x+3
We can see that
a=-1, b=-6, and
c=3. Now, we will follow four steps to graph the function.
- Find the .
- Calculate the .
- Identify the and its across the axis of symmetry.
- Connect the points with a .
Finding the Axis of Symmetry
The axis of symmetry is a with equation
x=-2ab. Since we already know the values of
a and
b, we can substitute them into the formula.
The axis of symmetry of the parabola is the vertical line with equation
x=-3.
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: (-2ab,f(-2ab))
Note that the formula for the
x-coordinate is the same as the formula for the axis of symmetry, which is
x=-3. Thus, the
x-coordinate of the vertex is also
-3. To find the
y-coordinate, we need to substitute
-3 for
x in the given equation.
f(x)=-x2−6x+3
f(-3)=-(-3)2−6(-3)+3
f(-3)=-(9)−6(-3)+3
f(-3)=-9+18+3
f(-3)=12
We found the
y-coordinate, and now we know that the vertex is
(-3,12).
Identifying the y-intercept and its Reflection
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 (0,3). Let's plot this point and its reflection across the axis of symmetry.
Connecting the Points
We can now draw the graph of the function. Since a=-1, which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.