Answer:
Step-by-step explanation:
From the given equations of the function,
1). f(x) = x² + 4x + 3
= x² + 2(2x) + 4 - 1
f(x) = (x + 2)² - 1
This is in the vertex form of a parabola [y = (x - h)² + k]
Here, (h, k) is the vertex of the parabola.
By comparing both the equations of the parabola,
(-2, 1) will be the vertex.
Table for input - output values,
x intercepts → (x + 2)² + 1 = 0
(x + 2) = ±1
x = -2 ± 1
x = -3, -1
Line of symmetry → x = -3
y-intercept of the graph, x = 0
y = (0 + 2)²- 1
= 4 - 1
y = 3
2). f(x) = x² - 6x + 11
= x² - 2(3x) + 9 + 2
= (x - 3)² + 2
By comparing the equation of the function with the vertex form of the parabola,
(3, -2) is the vertex.
Line of symmetry → x = 3
x-intercept → (x - 3)² + 2 = 0
(x - 3) = ±√(-2)
x = 3 ± √(-2) [Imaginary number]
Therefore, NO y-intercept.
y-intercept → y = 0 - 6(0) + 11 = 11
3). f(x) = -x² + 2x - 2
= -[x² - 2x + 2]
= -[x² - 2(1.x) + 1 - 1] - 2
= -[(x - 1)²- 1] - 2
= -(x - 1)²- 1
By comparing this equation with the vertex form of the equation,
(1, -1) is the vertex.
x - intercepts → y = -(x - 1)²- 1 = 0
(x - 1) = ±√(-1)
x = 1 ± √(-1) [Imaginary numbers]
Therefore, no x-intercepts.
y-intercept → y = -(0 - 1)² + 3
y = 2
4). y = [tex]\frac{1}{2}x^{2}-4x+5[/tex]
y = [tex]\frac{1}{2}(x^{2}-8x+10)[/tex]
y = [tex]\frac{1}{2}[x^{2}-2(4x)+10][/tex]
y = [tex]\frac{1}{2}[x^{2}-2(4x)+16-6][/tex]
y = [tex]\frac{1}{2}[(x-4)^2-6][/tex]
y = [tex]\frac{1}{2}(x-4)^2-3[/tex]
Vertex → (4, -3)
Line of symmetry → x = 4
x - intercepts → x = 4 ± √6
x = 1.55, 6.45
y - intercepts → y = 5
