Final answer:
To convert the quadratic function to vertex form, find the vertex using the formula h=-b/(2a), find the y-coordinate of the vertex by substituting h, determine the intercepts by setting x and g(x) to zero, identify the minimum point since a>0, and finally, describe the range as all y values greater than or equal to the y-coordinate of the vertex.
Step-by-step explanation:
First, we need to convert the quadratic function g(x)=0.15x²-0.3x-7 to vertex form, which is y=a(x-h)²+k, where (h,k) is the vertex of the parabola. To find the vertex, we can use the formula h=-b/(2a), where a=0.15 and b=-0.3. The vertex's x-coordinate is h=-(-0.3)/(2*0.15)=1. To find the y-coordinate of the vertex, we substitute h into the original equation, getting k=g(1).
Next, we find the intercepts by setting x and g(x) to zero separately. The y-intercept is obtained straight from the equation by setting x=0, which gives us the point (0,-7). The x-intercepts, or roots, are found by using the quadratic formula or factoring.
For maximum or minimum, because a>0, our parabola opens upwards, implying that the vertex is the minimum point. Lastly, the range of the function is all y values that the function can take, which, in this case, are all values y>=k, as the parabola has a minimum point and opens upwards.