A line y=c reflects the quadratic y=x2−4x+7 to yield y=−x2+4x+1. Find c and the vertex of the new parabola.
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Step-by-Step Solution
Step 1: Find the vertex of the original parabola
To find the vertex of a parabola in the form y=ax2+bx+d, we use the formula for the x -coordinate of the vertex, xv=−2ab. For the given parabola y=x2−4x+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 xv=−2(1)−4=2. Then, substitute xv=2 back into the original equation to find the y -coordinate of the vertex: yv=(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=−x2+4x+1, we have a=−1 and b=4. The x -coordinate of its vertex is xv′=−2(−1)4=2. Substituting xv′=2 into the new equation gives yv′=−(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=23+5=28=4.