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Understanding the vertex of a parabola is critical when analyzing quadratic functions. The vertex is the highest or lowest point on a parabola, depending on whether the parabola opens upwards or downwards. The vertex represents the turning point where the function changes direction.

For the quadratic function in vertex form, such as \(f(x)=a(x-h)^2+k\), the vertex is located at the point \((h,k)\). Here, \(h\) and \(k\) are constants that determine the horizontal and vertical displacement, respectively. In our example, \(f(x)=0.4(x-1)^2\), there is no vertical displacement, indicating that \(k=0\). So, the vertex of this parabola is at \((1,0)\). The significance of the vertex lies in its utility: it gives us a reference point from which we can graph the entire parabola and determine its key properties such as the axis of symmetry and the maximum or minimum value of the function.

The axis of symmetry in the context of a quadratic function is a vertical line that passes through the vertex of the parabola and divides it into two mirror-image halves. This line represents a fundamental symmetry for parabolas, implying that for every point on one side of the parabola, there is a corresponding point on the other side equidistant from the axis.

The equation for the axis of symmetry can be easily found from the vertex form of a quadratic function, \(f(x)=a(x-h)^2+k\), as \(x=h\). Taking our given function, \(f(x)=0.4(x-1)^2\), the axis of symmetry is at \(x=1\), exactly where the vertex lies.

Why is the axis of symmetry important?

It's not just a line to reflect the shape of the parabola; it's a tool that guides in graphing, helps to determine the domain of the function, and provides a method to find the roots of the quadratic equation.

The roots of a quadratic equation, also known as zeroes or x-intercepts, are the values of x at which the quadratic function \(f(x)\) equals zero. For the standard form of a quadratic equation \(ax^2+bx+c=0\), the roots can be found using various methods such as factoring, completing the square, or applying the quadratic formula.

For our quadratic function \(f(x)=0.4(x-1)^2\), you can find the roots by setting the equation equal to zero and solving for x. As described in the solution, setting \(0.4(x-1)^2 = 0\) leads to \((x-1)^2 = 0\) upon simplifying, which yields a single root at x = 1. This is an example of a parabola that touches the x-axis at just one point, the vertex, and thus has a single real root, also known as a double root since it can be considered twice under the same value.

Significance of Quadratic Roots

Knowing the roots of a quadratic equation is essential for graphing the function and for understanding its behavior, as they mark the points where the graph crosses or touches the x-axis.