Given:
f(x)=-x^2+4x+4f(x)=−x2+4x+4 is a quadratic equation in standard form:
f(x)=ax^2+bx+cf(x)=ax2+bx+c,
where:
a=-1a=−1, b=4b=4, and c=4c=4.
Vertex: maximum or minimum point of a parabola
The xx value of the vertex is the same as the axis of symmetry that divides the parabola into two equal halves.
x=(-b)/(2a)x=−b2a
x=(-4)/(2*-1)x=−42⋅−1
x=(-4)/(-2)x=−4−2 larr← Two negatives make a positive.
x=2x=2
To determine the yy value of the vertex, substitute yy for f(x)f(x). Substitute 22 for xx and solve for yy.
y=-(2)^2+4(2)+4y=−(2)2+4(2)+4
Simplify.
y=-4+8+4y=−4+8+4
y=8y=8
The vertex is (2,8)(2,8).
Since a<0a<0, the vertex is the maximum point and the parabola opens downward.
X-Intercepts: value of xx when y=0y=0
Substitute 00 for f(x)f(x) and solve for xx using the quadratic formula.
0=-x^2+4x+40=−x2+4x+4
Quadratic formula
x=(-b+-sqrt(b^2-4ac))/(2a)x=−b±√b2−4ac2a
Plug in the known values.
x=(-4+-sqrt((4)^2-4*-1*4))/(2*-1)x=−4±√(4)2−4⋅−1⋅42⋅−1
Simplify.
x=(-4+-sqrt(16+16))/(-2)x=−4±√16+16−2
Simplify 16+1616+16 to 3232.
x=(-4+-sqrt32)/(-2)x=−4±√32−2
Prime factorize 3232.
x=(-4+-sqrt((2xx2)xx(2xx2)xx2))/(-2)x=−4±√(2×2)×(2×2)×2−2
Simplify.
x=(-4+-2xx2sqrt2)/(-2)x=−4±2×2√2−2
x=(-4+-4sqrt2)/(-2)x=−4±4√2−2
Factor out the common 22.
x=(color(red)cancel(color(black)(-4))^2+-color(red)cancel(color(black)(4))^2sqrt2)/(color(red)cancel(color(black)(-2))^1
x=(2+-2sqrt2)
Solutions for x.
x=2+2sqrt2,2-2sqrt2
Simplify.
x=2(1+sqrt2),2(1-sqrt2)
x-intercepts: (2(1+sqrt2)),0), (2(1-sqrt2)),0)
Approximate intercepts: (-0.828,0),(4.83,0)
Y-Intercept: value of y when x=0.
Substitute 0 for x and solve for y.
y=-(0^2)+4(0)+4
y=4
y-intercept: (0,4)
Plot the points and sketch a parabola through the dots. Do not connect the dots.
graph{y=-x^2+4x+4 [-16.02, 16.02, -8.01, 8.01]}
