Given the equation y = x^2 - 16x + 89 with solutions of x = 8 - 5i and x = 8 + 5i, which of the following identifies the general shape of its associated graph?
the graph opens downward
the vertex is to the left of the y-axis
the graph touches the x-axis exactly twice
the graph lies above the x-axis
The correct answer is:
the graph lies above the x-axis
We can determine this by analyzing the solutions given. The solutions x = 8 - 5i and x = 8 + 5i are complex conjugates, which means that they occur in pairs, where the real part (8) is the same and only the imaginary part (±5i) differs in sign.
In this case, we have two solutions with a real part of 8 and an imaginary part of ±5i. When we analyze the imaginary parts, we see that they are ±5, which are non-zero values. This indicates that the solutions do not intersect the x-axis, which means that the graph lies above the x-axis.
To determine the general shape of the graph of the equation y = x^2 - 16x + 89, we can analyze the coefficients of the equation.
The coefficient of x^2 is positive (1), indicating that the graph opens upward. Therefore, the statement "the graph opens downward" is incorrect.
To find the vertex of the parabola, we can use the formula x = -b / (2a), where a and b are the coefficients of the equation. In this case, a = 1 and b = -16.
Using the formula, the x-coordinate of the vertex is x = -(-16) / (2*1) = 8.
Since the x-coordinate of the vertex is positive, the statement "the vertex is to the left of the y-axis" is incorrect.
To determine the number of times the graph touches the x-axis, we can look at the solutions given. The solutions x = 8 - 5i and x = 8 + 5i indicate that the graph does not intersect the x-axis at real number points. Therefore, the statement "the graph touches the x-axis exactly twice" is incorrect.
Based on the information provided, we can conclude that the statement "the graph lies above the x-axis" is correct.
To determine the general shape of the graph of the equation y = x^2 - 16x + 89, we can analyze the quadratic equation. The equation is in the form of y = ax^2 + bx + c, where a, b, and c are constants.
In this case, a = 1, b = -16, and c = 89. Since a is positive (a = 1), the graph opens upward, not downward, eliminating the first option.
To find the vertex of the parabola, we can use the formula x = -b/2a. In this case, substituting the values of a and b, we get x = 16/2 = 8. The x-coordinate of the vertex is 8. Since it is positive, the vertex is indeed to the right of the y-axis, eliminating the second option.
Next, to determine the number of times the graph touches the x-axis, we can look at the solutions of the equation. Given that x = 8 - 5i and x = 8 + 5i are the solutions, it means that the quadratic does not intersect the x-axis at any real points. Instead, it intersects the x-axis at two complex points. Therefore, the graph does not touch the x-axis exactly twice, eliminating the third option.
Finally, since the equation is a quadratic with a positive leading coefficient (a = 1), the graph lies above the x-axis for all values of x. Hence, the correct answer is that the graph lies above the x-axis, which is the fourth option.
In summary, the correct answer is: "The graph lies above the x-axis."