Find the vertex, x and y-intercepts, and sketch a graph

Question

𝑓𝑥 =−2𝑥^2−8𝑥+1

Answer 1

f(x) = -2x² - 8x + 1

 

VERTEX      If A = -2, B = -8, and C = 1,  the x-component of the vertex is -B/2A  = +8/-4 = -2.   To find the y-component of the vertex, substitute -2 for x.  -2(-2)²-8(-2)+1 = -8+16+1 = 9

So the vertex is (-2, 9)

 

X-INTERCEPTS     The x intercepts occur where the graph crosses the y-axis.  That is, where y=0.

0 = -2x² - 8x + 1  This can be solved by several methods; graphing, factoring, completing the square or the quadratic formula.  Using the A, B, C from above and the formula x = -B ± sqrt (B² - 4AC) All over 2A

x = 8 ± sqrt [64 - 4 ( -2) * 1] All over 2 (-2)

x = 8 ± sqrt[72]  All over -4

So the x-intercepts are [8 + sqrt (72)]/-4 and [8 - sqrt (72)]/-4   Or approximately (8 + 8.485)/-4 = -4.1213 and         (8 - 8.485)/-4 = .1213

 

Y-INTERCEPTS    The y-intercepts occur when x = 0.  So just substitute 0 in the equation and get y = 1.

 

Graph - This is a downward opening parabola.  It goes through the 4 points you found:  (-2,9), (-4.1213, 0) and    (.1213, 0) and (0,1)