Question

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graph the equation.
y = 2(x - 4)² + 5

Explanation:

Step1: Identify vertex form

The quadratic equation is in vertex form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex. For $y=2(x-4)^2+5$, $h=4$, $k=5$, so vertex is $(4,5)$.

Step2: Find vertex point

Plot the vertex $(4,5)$ (this is the green point on the graph).

Step3: Determine direction/width

Since $a=2>0$, parabola opens upward. $|a|>1$, so it is narrower than $y=x^2$.

Step4: Find additional points

Choose $x=5$: $y=2(5-4)^2+5=2(1)+5=7$, so point $(5,7)$.
Choose $x=3$: $y=2(3-4)^2+5=2(1)+5=7$, so point $(3,7)$.
Choose $x=6$: $y=2(6-4)^2+5=2(4)+5=13$, so point $(6,13)$.
Choose $x=2$: $y=2(2-4)^2+5=2(4)+5=13$, so point $(2,13)$.

Step5: Plot & connect points

Plot the symmetric points and draw a smooth upward-opening parabola with vertex at $(4,5)$.

Answer:

The graph is an upward-opening parabola with vertex at $(4,5)$, passing through symmetric points such as $(3,7)$, $(5,7)$, $(2,13)$, $(6,13)$, and following the curve shown (with the vertex at the green point $(4,5)$ and extending upward symmetrically on both sides of $x=4$).