Question

y=–x2+12x–4

Answer 1

[AI-generated answer — Bot GPT 3.5]

The given equation is a quadratic equation in the standard form.

To analyze this equation, we can first determine the vertex of the parabola represented by the equation.

The x-coordinate of the vertex is given by -b/2a, where a and b are the coefficients of the quadratic equation. In this case, a = -1 and b = 12.

x-coordinate of the vertex = -12/ (2 * -1) = -12 / -2 = 6

To find the y-coordinate of the vertex, we substitute the x-coordinate into the equation.

y = -6^2 + 12(6) - 4 = -36 + 72 - 4 = 32

Therefore, the vertex of the parabola is (6, 32).

The parabola opens downwards because the coefficient of the x^2 term is negative. The vertex is the highest point.

Now, we can determine the axis of symmetry, which is a vertical line passing through the vertex. The equation of the axis of symmetry is x = 6.

Finally, we can find the y-intercept by substituting x = 0 into the equation.

y = -0^2 + 12(0) - 4 = 0 + 0 - 4 = -4

Therefore, the y-intercept is -4.

In summary, the vertex of the parabola is (6, 32), the axis of symmetry is x = 6, and the y-intercept is -4.

Answer 2

[AI-generated answer — Explain Bot]

The given equation is y = -x^2 + 12x - 4.

To understand the characteristics of this equation, we can analyze its components:

1. The coefficient of x^2 is -1, which means the graph of the equation is a downward-opening parabola.

2. The coefficient of x is 12, indicating that the vertex of the parabola will have a horizontal coordinate of -b/2a = -12/(2*(-1)) = -12/(-2) = 6.

3. To find the vertical coordinate of the vertex, substitute the x-coordinate (6) back into the equation: y = -(6)^2 + 12(6) - 4 = -36 + 72 - 4 = 32.

Therefore, the vertex of this parabola is (6, 32).

Now we can determine some additional information about the parabola:

1. Axis of symmetry: Since the parabola is symmetrical, the axis of symmetry is the vertical line passing through the vertex, which is x = 6.

2. Y-intercept: To find the y-intercept, substitute x = 0 into the equation: y = -(0)^2 + 12(0) - 4 = 0 + 0 - 4 = -4. Thus, the y-intercept is (0, -4).

3. X-intercept(s): To find the x-intercept(s), set y = 0 in the equation and solve for x. However, in this case, the equation is a quadratic that doesn't factor easily. We can use the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a).

Using a = -1, b = 12, and c = -4, we have:
x = (-12 ± √(12^2 - 4*(-1)*(-4)))/(2*(-1))
x = (-12 ± √(144 - 16))/(-2)
x = (-12 ± √128)/(-2)
x = (-12 ± 8√2)/(-2)
x = 6 ± 4√2

Therefore, the x-intercepts are (6 + 4√2, 0) and (6 - 4√2, 0).

Now, armed with this information, we can plot the parabola and its key points on a graph.

Answer 3

[AI-generated answer — Step-by-Step Bot]

To graph the equation y = -x^2 + 12x - 4, we can follow these steps:

Step 1: Find the vertex of the parabola.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a and b are the coefficients of x^2 and x respectively.
In this case, a = -1 and b = 12.
x = -12 / (2*(-1)) = -12 / -2 = 6.
To find the y-coordinate of the vertex, substitute the x-coordinate into the equation:
y = -(6)^2 + 12(6) - 4 = -36 + 72 - 4 = 32.
So, the vertex of the parabola is (6, 32).

Step 2: Find the y-intercept.
To find the y-intercept, substitute x = 0 into the equation:
y = -(0)^2 + 12(0) - 4 = 0 + 0 - 4 = -4.
So, the y-intercept is (0, -4).

Step 3: Find the x-intercepts.
To find the x-intercepts, set y = 0 in the equation and solve for x:
0 = -x^2 + 12x - 4.
This equation does not factor nicely, so we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = -1, b = 12, and c = -4.
x = (-12 ± √(12^2 - 4*(-1)*(-4))) / (2*(-1)).
x = (-12 ± √(144 - 16)) / (-2).
x = (-12 ± √128) / (-2).
x = (-12 ± 8√2) / (-2).
Simplifying further, x = 6 ± 4√2.
So, the x-intercepts are (6 + 4√2, 0) and (6 - 4√2, 0).

Step 4: Plot the points and sketch the parabola.
Now that we have the vertex, y-intercept, and x-intercepts, we can plot these points and sketch the parabola using a graphing tool or by hand.

The vertex is (6, 32). The y-intercept is (0, -4). The x-intercepts are approximately (12.7, 0) and (-0.7, 0).

The graph of the equation y = -x^2 + 12x - 4 is a downward-opening parabola that passes through these points (vertex, y-intercept, x-intercepts).

Note: The graph may vary depending on the scale of the axes and accuracy of the x-intercept values.