To identify which graph represents the function f(x) = x3 + x2 + 10x + 8, we need to analyze the characteristics of the function, such as its shape, intercepts, and behavior at infinity.
This function is a polynomial of degree 3, meaning its leading coefficient and degree will dictate its end behavior. As x approaches positive infinity, f(x) will also approach positive infinity, and as x approaches negative infinity, f(x) will approach negative infinity. Therefore, we expect the graph to rise to the right and fall to the left.
Next, we can determine the y-intercept by evaluating f(0):
- f(0) = 03 + 02 + 10(0) + 8 = 8.
This tells us that the graph crosses the y-axis at (0, 8).
To find the x-intercepts, we would need to solve the equation f(x) = 0. This can be done through numerical methods or graphing calculators, as the cubic equation may not yield simple factors. The real roots will indicate where the function crosses the x-axis.
We may also want to check the derivatives to find critical points, which could help in identifying local maxima or minima. By calculating the first derivative f'(x), we can find where the slope of the function equals zero:
- f'(x) = 3x2 + 2x + 10.
Solve f'(x) = 0 to find critical points, where we’ll evaluate changes in concavity and determine if these points represent a maximum or minimum. Since the quadratic formula will tell us about the roots, we can confirm that this expression always yields positive values (given that the discriminant is negative), indicating there are no real critical points.
Once we have plotted the function and analyzed its general behavior, we can compare our findings to the provided graphs. The correct graph will show the characteristics we’ve determined: it will pass through (0, 8), extend towards positive values as x increases and approach negative values as x decreases, and it will be smoothly curved given that it is a cubic function.
In summary, to select the right graph for f(x) = x3 + x2 + 10x + 8, look for a curve that adheres to these characteristics: two ends going to opposites (upward to the right, downward to the left), a y-intercept at (0, 8), and no x-intercepts where it curves without flat points.