Question

Jermaine is studying a quadratic function: he finds the function has only one root, -4.

Which statements below must be true about the graph of Jermaine's function?
I. his parabola must open don
II. the vertex of his parabola must be (-4, 0)
III. the x-intercept of his parabola mus be (-4, 0)
IV. the axis of symmetry of his parabola must be x = -4

A. I and Il only
B. III and IV only
C. 11, IIl and IV only
D. 1, 11, IIl and IV

Answer 1

The correct statements about the graph of Jermaine's quadratic function with one root at -4 are that the vertex must be (-4, 0), the x-intercept must be (-4, 0), and the axis of symmetry must be x = -4. The direction in which the parabola opens cannot be deduced from this information.

Step-by-step explanation:

If Jermaine finds that a quadratic function has only one root, which is -4, certain statements about the graph of this function can be made. When a quadratic has one root, it means that the graph of the function touches the x-axis at one point, known as a double root. Here are the statements:

• Statement I: This statement is incorrect. A parabola can open either upwards or downwards and still have only one root if the vertex is on the x-axis.
• Statement II: The vertex of his parabola must be (-4, 0). This statement is true since, in a quadratic function, if there's only one root, it means that the parabola's vertex lies on the x-axis at the point where x equals the root.
• Statement III: The x-intercept of his parabola must be (-4, 0). This is true as well, since the root of a quadratic function is the x-value where the function equals zero, which also corresponds to the x-intercept.
• Statement IV: The axis of symmetry of his parabola must be x = -4. This statement is also true because, in a parabola, the axis of symmetry always passes through the vertex.

Considering the above explanations, the correct answer is C) II, III, and IV only. Statement I cannot be determined solely based on the information about the root.