Graph the function 𝑓(𝑥) = −2𝑥2 + 2𝑥 + 4 and its reciprocal on the same or separateaxes. Explain your thinking

To graph the function 𝑓(𝑥) = −2𝑥2 + 2𝑥 + 4 and its reciprocal, follow these steps:

Step 1: Identify the type of function The given function is a quadratic function because it is in the form of ax^2 + bx + c.

Step 2: Identify the key features of the function The key features of a quadratic function are its vertex, axis of symmetry, and y-intercept.

For the given function, the vertex can be found using the formula -b/2a, which gives us x = 0.5. Substituting x = 0.5 into the function gives us the y-coordinate of the vertex, which is 3.5. So, the vertex is (0.5, 3.5).

The axis of symmetry is the vertical line passing through the vertex, which is x = 0.5.

The y-intercept is the value of the function when x = 0, which is 4.

Step 3: Sketch the function Plot the vertex, axis of symmetry, and y-intercept on a graph. Then, draw a parabola that passes through these points. The parabola opens downwards because the coefficient of x^2 is negative.

Step 4: Graph the reciprocal of the function The reciprocal of the function is 1/𝑓(𝑥) = 1/(-2𝑥2 + 2𝑥 + 4). To graph this function, plot points for a range of x-values and connect them. Note that the reciprocal function will not be defined where the original function is 0, and it will approach 0 where the original function approaches infinity.

Remember to label your graphs clearly and indicate any points of discontinuity on the graph of the reciprocal function.