Use any method to locate the zeros of y=−3x^2+12x+15 .
(0,15) left parenthesis 0 comma 15 right parenthesis (1,0)(−5,0) left parenthesis 1 comma 0 right parenthesis left parenthesis negative 5 comma 0 right parenthesis There are no zeros. There are no zeros. (−1,0)(5,0)
To locate the zeros of the equation y = -3x^2 + 12x + 15, we can set y equal to zero and solve for x:
0 = -3x^2 + 12x + 15
Next, we can factor the equation or use the quadratic formula to find the zeros.
Factoring:
0 = -3(x^2 - 4x - 5)
0 = -3(x - 5)(x + 1)
Setting each factor equal to zero:
x - 5 = 0 --> x = 5
x + 1 = 0 --> x = -1
Therefore, the zeros of the equation are x = 5 and x = -1.
Complete the square to identify the vertex of y=x^2−16x+14 .
(8,78) left parenthesis 8 comma 78 right parenthesis (8,−50) left parenthesis 8 comma negative 50 right parenthesis (8,14) left parenthesis 8 comma 14 right parenthesis (8,30)
To complete the square for the equation y = x^2 - 16x + 14, we can follow these steps:
1. Group the x terms together: y = (x^2 - 16x) + 14.
2. Complete the square in the parentheses: y = (x^2 - 16x + 64) - 64 + 14.
3. Simplify the inside of the parentheses: y = (x - 8)^2 - 50.
Now we can see that the equation is in the form y = a(x - h)^2 + k, where (h, k) is the vertex.
Therefore, the vertex of the equation y = x^2 - 16x + 14 is (8, -50).
Identify the vertex of y=−10(x+15)^2−18 .
(−150,−18) left parenthesis negative 150 comma negative 18 right parenthesis (−10,−15) left parenthesis negative 10 comma negative 15 right parenthesis (−15,−18) left parenthesis negative 15 comma negative 18 right parenthesis (15,−18)
To identify the vertex of the equation y = -10(x + 15)^2 - 18, we can compare it to the standard form of a quadratic equation y = a(x - h)^2 + k. The vertex of a quadratic function is given by the coordinates (h, k).
In this case, we have:
a = -10
h = -15
k = -18
So, the vertex is (-15, -18).
Create a graph; then answer the following question. What are the key points on the graph of y=x^2−8x ? Name the vertex, x-intercept(s), and y-intercept.
x-intercept: (0,8) y-intercept: (0,0) vertex: (4,−16) x -intercept: left parenthesis 0 comma 8 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 4 comma negative 16 right parenthesis x-intercepts: (0,0) (8,0) y-intercept: (0,0) vertex: (4,−16) x -intercepts: left parenthesis 0 comma 0 right parenthesis left parenthesis 8 comma 0 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 4 comma negative 16 right parenthesis x-intercept: (4,−16) y-intercept: (0,0) vertex: (8,0) x -intercept: left parenthesis 4 comma negative 16 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 8 comma 0 right parenthesis x-intercepts: (0,0) (0,8) vertex: (4,−16) There is no y-intercept.
To create a graph of the equation y = x^2 - 8x, we can plot the key points and connect them to form a parabola.
The key points are:
x-intercept: (0,0)
y-intercept: (0,0)
vertex: (4,-16)
Now let's plot these points on a graph:
The x-intercept (0,0) is where the graph crosses the x-axis.
The y-intercept (0,0) is where the graph crosses the y-axis.
The vertex (4,-16) is the lowest point on the graph.
Here is the graph of the equation y = x^2 - 8x:
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As shown in the graph, the vertex is at (4,-16), the x-intercepts are at (0,0) and (8,0), and the y-intercept is at (0,0).
What point is symmetrical to (−1,−17) for the graph of y=−x^2−16 ?
(1,17) left parenthesis 1 comma 17 right parenthesis (1,−17) left parenthesis 1 comma negative 17 right parenthesis (−1,17) left parenthesis negative 1 comma 17 right parenthesis (−17,−1) left parenthesis negative 17 comma negative 1 right parenthesis
To find the point that is symmetrical to (-1,-17) for the graph of y = -x^2 - 16, we can use the concept of symmetry.
The graph of a quadratic function is symmetric with respect to the vertical line passing through its vertex. In this case, the vertex of the quadratic function y = -x^2 - 16 is (-1,-16).
To find the point that is symmetrical to (-1,-17), we need to find the point that has the same x-coordinate but lies on the other side of the vertex. Since the x-coordinate is -1, the point on the other side of the vertex will also have an x-coordinate of -1.
The y-coordinate of the symmetrical point can be found by reflecting the y-coordinate of (-1,-17) with respect to the line of symmetry. The line of symmetry is the vertical line passing through the vertex.
So, the symmetrical point to (-1,-17) is (-1,17).
To locate the zeros of the given equation y = -3x^2 + 12x + 15, we need to find the values of x for which y becomes zero. In other words, we need to solve the equation -3x^2 + 12x + 15 = 0.
There are several methods to solve this quadratic equation, including factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula in this case.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.
For the given equation, the coefficients are:
a = -3
b = 12
c = 15
Substituting these values into the quadratic formula, we get:
x = (-12 ± √(12^2 - 4(-3)(15))) / (2*(-3))
Simplifying further:
x = (-12 ± √(144 + 180)) / (-6)
x = (-12 ± √324) / (-6)
x = (-12 ± 18) / (-6)
Now, we can find the values for x by evaluating both possibilities of the ± sign:
For x = (-12 + 18) / (-6):
x = 6 / (-6)
x = -1
For x = (-12 - 18) / (-6):
x = -30 / (-6)
x = 5
Therefore, the zeros of the equation y = -3x^2 + 12x + 15 are x = -1 and x = 5.