Graph the equation and its tangent
-$$\text { Graph } y = x ^ { 3 } - 4 \text { and the tangent to the curve at the point whose } x \text {-coordinate is } 0 \text {. }$$

Question

Quizplus · Accepted Answer

To graph the equation, we can use a table of values or plot points. For simplicity, we can choose x values from -3 to 3.

| x | y |
|---|---|
| -3 | -31 |
| -2 |-12 |
| -1 | -5 |
|  0 | -4 |
|  1 | -3 |
|  2 |  4 |
|  3 | 23 |

Plotting these points, we get:

11ece0ba_7c5b_690f_8008_2ba1e37e0507_TB8583_11

To find the equation of the tangent at the point (0,-4), we need to find its slope first. The slope of a tangent is equal to the derivative of the function at the given point. 
$$y=x^3-4 $$
$$\frac{dy}{dx}=3x^2$$
At x=0, 
$$\frac{dy}{dx} = 3 \times 0^2 = 0$$
Therefore, the slope of the tangent is 0. The equation of the tangent is of the form y=c where c is a constant. Since the tangent passes through (0,-4), c=-4. Therefore, the equation of the tangent is y=-4.

Plotting the tangent on the same graph, we get:

11ece0ba_6983_63db_8008_67de33627e51_TB8583_11

Therefore, the best choice is B.