Question

a companys logo
a company’s logo is represented in the figure on the right.
this logo is bounded by the horizontal line ( l ), a parabola representing function ( f ), and a parabola representing function ( g ).
the two parabolas intersect on the ( x )-axis at points ( a ) and ( b ).
the line ( l ) and the parabola representing function ( f ) intersect at points ( c ) and ( d ).
point ( c ) is located on the ( y )-axis.
the vertex of the parabola representing function ( g ) is ( v(6, 2) ).
the ( y )-intercept of function ( f ) is ( 5 ).
the coordinates of point ( b ) are ( b(10, 0) ).
determine the equations of the four borders of the logo.

Explanation:

Step1: Find parabola g's equation

Vertex form: $y = a(x-h)^2 + k$, where $(h,k)=(6,2)$. Substitute $B(10,0)$:
$0 = a(10-6)^2 + 2$
$0 = 16a + 2$
$a = -\frac{2}{16} = -\frac{1}{8}$
Equation: $y = -\frac{1}{8}(x-6)^2 + 2$
Expand: $y = -\frac{1}{8}(x^2-12x+36) + 2 = -\frac{1}{8}x^2 + \frac{3}{2}x - \frac{9}{2} + 2 = -\frac{1}{8}x^2 + \frac{3}{2}x - \frac{5}{2}$
Find point A: set $y=0$
$0 = -\frac{1}{8}(x-6)^2 + 2$
$(x-6)^2 = 16$
$x-6 = \pm4$
$x=10$ (point B) or $x=2$, so $A(2,0)$

Step2: Find line l's equation

Line l is horizontal, passes through $C(0,5)$, so $y=5$

Step3: Find parabola f's equation

Parabola f passes through $A(2,0)$, $B(10,0)$, y-intercept $(0,5)$. Use standard form $y = ax^2 + bx + c$:
$c=5$
Substitute $A(2,0)$: $0 = 4a + 2b + 5$
Substitute $B(10,0)$: $0 = 100a + 10b + 5$
Solve system:
Equation1: $4a + 2b = -5$
Equation2: $100a + 10b = -5$ → $10a + b = -\frac{1}{2}$
Multiply Equation2 by 2: $20a + 2b = -1$
Subtract Equation1: $16a = 4$ → $a = \frac{1}{4}$
Substitute $a=\frac{1}{4}$ into $10a + b = -\frac{1}{2}$:
$10(\frac{1}{4}) + b = -\frac{1}{2}$ → $\frac{5}{2} + b = -\frac{1}{2}$ → $b = -3$
Equation: $y = \frac{1}{4}x^2 - 3x + 5$

Step4: Find point D's coordinates

Point D is intersection of line $l: y=5$ and parabola f:
$5 = \frac{1}{4}x^2 - 3x + 5$
$\frac{1}{4}x^2 - 3x = 0$
$x(\frac{1}{4}x - 3) = 0$
$x=0$ (point C) or $x=12$, so $D(12,5)$

Answer:

1. Equation of line $l$: $\boldsymbol{y=5}$
2. Equation of parabola $g$: $\boldsymbol{y = -\frac{1}{8}(x-6)^2 + 2}$ (or expanded: $\boldsymbol{y = -\frac{1}{8}x^2 + \frac{3}{2}x - \frac{5}{2}}$)
3. Equation of parabola $f$: $\boldsymbol{y = \frac{1}{4}x^2 - 3x + 5}$
4. The four borders are defined by:

• Horizontal line: $\boldsymbol{y=5}$
• Parabola $f$: $\boldsymbol{y = \frac{1}{4}x^2 - 3x + 5}$
• Parabola $g$: $\boldsymbol{y = -\frac{1}{8}(x-6)^2 + 2}$
• The segment of the x-axis between $A(2,0)$ and $B(10,0)$ (implicit border of the logo shape)