Similarly, for the new parabola y = -x² + 4x + 1, we have a = -1 and b = 4. The x -coordinate of its vertex is x_v' = -(4)/(2(-1)) = 2. Substituting x_v' = 2 into the new equation gives y_v' = -(2)² + 4(2) + 1 = -4 + 8 + 1 = 5. So, the vertex of the new parabola is (2, 5).

Similarly, for the new parabola y = -x² + 4x + 1, we have a = -1 and b = 4. The x -coordinate of its vertex is x_v' = -(4)/(2(-1)) = 2. Substituting x_v' = 2 into the new equation gives y_v' = -(2)² + 4(2) + 1 = -4 + 8 + 1 = 5. So, the vertex of the new parabola is (2, 5).

A line $y = c$ reflects the quadratic $y = x^2 - 4x + 7$ to yield $y = -x^2 + 4x + 1$ . Find $c$ and the vertex of the new parabola.

Answer: The value of $c$ is $4$ and the vertex of the new parabola is $(2, 5)$ .

Step-by-step solution

Step 1: Find the vertex of the original parabola

To find the vertex of a parabola in the form $y = ax^2 + bx + d$ , we use the formula for the $x$ -coordinate of the vertex, $x_v = -\frac{b}{2a}$ . For the given parabola $y = x^2 - 4x + 7$ , we have $a = 1$ and $b = -4$ .

Step 2: Calculate the vertex of the original parabola

Substitute $a = 1$ and $b = -4$ into the formula to get $x_v = -\frac{-4}{2(1)} = 2$ . Then, substitute $x_v = 2$ back into the original equation to find the $y$ -coordinate of the vertex: $y_v = (2)^2 - 4(2) + 7 = 4 - 8 + 7 = 3$ . So, the vertex of the original parabola is $(2, 3)$ .

Step 3: Find the vertex of the new parabola

Similarly, for the new parabola $y = -x^2 + 4x + 1$ , we have $a = -1$ and $b = 4$ . The $x$ -coordinate of its vertex is $x_v' = -\frac{4}{2(-1)} = 2$ . Substituting $x_v' = 2$ into the new equation gives $y_v' = -(2)^2 + 4(2) + 1 = -4 + 8 + 1 = 5$ . So, the vertex of the new parabola is $(2, 5)$ .

Step 4: Determine the line of reflection

When a parabola is reflected across a horizontal line $y = c$ , the $x$ -coordinate of its vertex remains the same, while the $y$ -coordinate changes. The line of reflection $y = c$ is exactly halfway between the $y$ -coordinates of the original vertex and the new vertex. Therefore, $c$ is the midpoint of the $y$ -coordinates of the two vertices.

Step 5: Calculate the value of c

Using the $y$ -coordinates of the original vertex $(2, 3)$ and the new vertex $(2, 5)$ , we calculate $c = \frac{3 + 5}{2} = \frac{8}{2} = 4$ .

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A line y=cy = cy=c reflects the quadratic y=x2−4x+7y = x^2 - 4x + 7y=x2−4x+7 to yield y=−x2+4x+1y = -x^2 + 4x + 1y=−x2+4x+1. Find ccc and the vertex of the new parabola.