The axis of symmetry of the parabola is the vertical line with equation x=3.
Calculating the Vertex
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=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.
We found the y-coordinate, and now we know that the vertex is (3,- 4).
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 (0,5). 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 positive, the parabola will open upwards. Let's connect the three points with a smooth curve.
We have already found that the vertex is (3, -4), which we identify on the above graph.
b The domain of quadratic functions is all real numbers. We can see on the graph in Part A that the minimum point of the curve is reached at the vertex. Thus, the range is all real numbers greater than or equal to - 4.
Domain:& All real numbers
Range:& y ≥ - 4
c Looking at the graph the given function, we can tell that the parabola opens upwards.
The vertex is always the lowest or the highest point on the graph. Therefore, in this case, the vertex represents the minimum value of the function.
