The given function is a quadratic function in standard form:
f(x) = -0.25x² + 8x - 2
To determine whether the function opens upwards or downwards, we can look at the coefficient of the x² term, which is negative (-0.25). Therefore, the function opens downwards.
The vertex of the function can be found using the formula:
x = -b / (2a)
where a and b are the coefficients of the x² and x terms, respectively. In this case, a = -0.25 and b = 8. Substituting these values into the formula, we get:
x = -8 / (2*(-0.25)) = 16
To find the y-coordinate of the vertex, we can substitute x = 16 into the function:
f(16) = -0.25(16)² + 8(16) - 2 = 62
Therefore, the vertex of the function is (16, 62).
The equation for the axis of symmetry is given by:
x = h
where h is the x-coordinate of the vertex. In this case, h = 16. Therefore, the equation for the axis of symmetry is:
x = 16
To sketch the graph of the function, we can use the vertex and the axis of symmetry to plot the point (16, 62) and draw the axis of symmetry as a vertical line at x = 16. We can also plot a few other points to get an idea of the shape of the graph. For example, we can substitute x = 0 and x = 32 into the function to get:
f(0) = -0.25(0)² + 8(0) - 2 = -2
f(32) = -0.25(32)² + 8(32) - 2 = 30
Plotting these points and connecting them with a smooth curve, we get the following sketch:
Quadratic function sketch
