Question

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=x^{2}-0.1 x $$

Short answer

The graph of the function \(f(x) = x^2 - 0.1x\) can be plotted in a viewing window with X-axis from -0.5 to 0.5 and Y-axis from -0.1 to 0.2. Important features include the vertex at (0.05, -0.0025), the y-intercept at (0, 0), and the x-intercepts at (0, 0) and (0.1, 0). Plot the vertex, intercepts, and sketch the parabolic shape within the chosen viewing window.

Step-by-step solution

Identify the function type and important features

The function is a quadratic function of the form \(f(x) = ax^2 + bx + c\), where a = 1, b = -0.1, and c = 0. We can identify some important features of the graph: 1. The vertex: We can find the vertex of the parabola by using the formula \(\displaystyle h=\dfrac{-b}{2a}\) for the x-coordinate and plugging in the value of h into the function to find the y-coordinate. Vertex: \(h = \dfrac{-(-0.1)}{2\cdot 1} = 0.05 , k = f(0.05)=0.05^2-0.1\cdot0.05=-\:0.0025\) Vertex: \((0.05, -0.0025)\) 2. The y-intercept: We can find the y-intercept by plugging in x = 0 into the function. Y-intercept: \(f(0) = 0^2 - 0.1 \cdot 0 = 0\) Y-intercept: (0, 0) 3. The x-intercepts: We can find the x-intercepts by setting the function equal to 0 and solving for x. \(x^2 - 0.1x = 0\) \(x(x - 0.1) = 0\) The x-intercepts are x = 0 and x = 0.1.

Determine a suitable viewing window

Given the important features such as the vertex and the intercepts, we can choose a suitable viewing window to see the overall shape of the parabola. Since the parabola has its vertex close to the origin and considering the x-intercepts, we may choose the viewing window as follows: - X-axis: From -0.5 to 0.5 (covers the x-intercepts and the vertex) - Y-axis: From -0.1 to 0.2 (covers the vertex and leaves some space above)

Plot the graph in the selected viewing window

Having chosen a suitable viewing window, we can now plot the graph of the function \(f(x) = x^2 - 0.1x\) within that window. We will use the grid with the X-axis ranging from -0.5 to 0.5 and the Y-axis ranging from -0.1 to 0.2. 1. Plot the vertex (0.05, -0.0025) 2. Plot the y-intercept (0, 0) 3. Plot the x-intercepts (0, 0) and (0.1, 0) 4. Use these points and the shape of a parabola to sketch the graph of the function. Once you've completed these steps, you would have a graph representing the function \(f(x) = x^2 - 0.1x\), correctly plotted within the viewing window.