Let's address each question step by step.
Question 1
To find the zeros of the quadratic equation \(y = -3x^2 + 12x + 15\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = -3\), \(b = 12\), and \(c = 15\).
First, calculate the discriminant:
\[ b^2 - 4ac = 12^2 - 4(-3)(15) = 144 + 180 = 324 \]
Now, apply the quadratic formula:
\[ x = \frac{-12 \pm \sqrt{324}}{2(-3)} = \frac{-12 \pm 18}{-6} \]
Calculating the two possible \(x\) values:
- \(x = \frac{6}{-6} = -1\)
- \(x = \frac{-30}{-6} = 5\)
So, the zeros are \((-1, 0)\) and \((5, 0)\).
Response: \((-1, 0), (5, 0)\)
Question 2
To complete the square for the equation \(y = x^2 - 16x + 14\):
- Take the coefficient of \(x\) (which is \(-16\)), divide by \(2\) to get \(-8\), and then square it to get \(64\).
- Rewrite the equation:
\[ y = (x^2 - 16x + 64) - 64 + 14 \]
This simplifies to:
\[ y = (x - 8)^2 - 50 \]
The vertex form \(y = (x - h)^2 + k\) shows that the vertex is at \((8, -50)\).
Response: \((8, -50)\)
Question 3
For the equation \(y = -10(x + 15)^2 - 18\), the vertex form is already presented clearly. The vertex \((h, k)\) is:
\((h, k) = (-15, -18)\).
Response: \((-15, -18)\)
Question 4
To find the key points of the graph \(y = x^2 - 8x\):
-
Vertex: Complete the square: \[ y = (x^2 - 8x + 16) - 16 = (x - 4)^2 - 16 \] Vertex is \((4, -16)\).
-
x-intercepts: Set \(y = 0\): \[ 0 = x^2 - 8x \rightarrow x(x - 8) = 0 \rightarrow x = 0 \text{ or } x = 8 \] x-intercepts are \((0, 0)\) and \((8, 0)\).
-
y-intercept: Set \(x = 0\): \[ y = 0^2 - 8(0) = 0 \quad \Rightarrow \quad (0, 0) \]
Response:
- x-intercepts: \((0, 0)\) and \((8, 0)\)
- vertex: \((4, -16)\)
- y-intercept: \((0, 0)\)
Question 5
To find the point symmetrical to \((-1, -17)\) across the graph of \(y = -x^2 - 16\), we need to reflect this point.
- Determine the symmetry line (axis of symmetry) based on the general form of the parabola. The vertex of this parabola is at \((0, -16)\), which acts as the line of symmetry.
- The y-values will change sign relative to the line \(y = -16\).
- The vertical distance from \(y = -17\) to \(y = -16\) is \(1\), so reflected across the line of symmetry gives \(y = -15\).
Thus, \((-1, -17)\) reflects to \((-1, -15)\).
However, observing the common choices: It suggests points symmetrical with respect to x-values only, so:
- The reflection would be to \( (-1, 17) \) since we only consider y-values being affected symmetrically if following a polynomial reflection.
Response: \((-1, 17)\)
If there are any issues or clarifications needed in the solutions, feel free to ask!
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