Answer:
no, the vertex of Function A is 6 units lower
Step-by-step explanation:
The vertex of Function A can be found several ways:
- putting the equation in vertex form
- evaluating the function on the axis of symmetry
- graphing the function
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Vertex Form
To put the equation in vertex form, we can consider separately the variabel terms and the constant. Once we have the variable terms separated, we can "complete the square" and then simplify the result to vertex form. The vertex form of a quadratic is ...
y = a(x -h)² +k . . . . for vertical scale factor 'a' and vertex (h, k)
We have ...
y = 2x² +12x +10
y = 2(x² +6x) +10
We complete the square by adding the square of half the x-coefficient inside parentheses. We subtract the same quantity outside parentheses, so we don't change the equation any.
y = 2(x² +6x +9) +10 -2(9)
y = 2(x +3)² -8 . . . . . . simplify to vertex form
The vertex of Function A is (-3, -8), which is (-2 -(-8)) = 6 units lower than the vertex of Function B in the graph. Bob is incorrect.
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Axis of Symmetry
The axis of symmetry of quadratic expression ax² +bx +c is x=-b/(2a). For the given Function A, the axis of symmetry is ...
x = -12/(2(2)) = -3
The height of the vertex is the y-value for that x-value:
y = 2(-3)² +12(-3) +10 = 18 -36 +10 = -8
This is lower than the vertex of the graphed function, which is at y=-2. Bob is incorrect.
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Graphing
The graph of Function A is the solid red curve in the attached. The graph of Function B is added as the dashed blue curve for reference. The graph clearly shows the vertex of Function A is lower than that of Function B. Bob is incorrect.
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Additional comment
Modern graphing calculators make the graphing option the quickest and easiest way to find the vertex of the function. The only "work" is to type the equation into the calculator.
