Answer: a) h = 2
b) a = [tex]-\dfrac{3}{4}[/tex] k = [tex]\bold{3\dfrac{1}{2}}[/tex]
Step-by-step explanation:
The vertex form of a quadratic equation is: y = a(x - h)² + k where
- a is the vertical stretch
- (h, k) is the vertex
The graph shows the axis of symmetry at x = 2, therefore, the x-coordinate of the vertex (h) is 2.
Two points are given: [tex]\bigg(0, \dfrac{1}{2}\bigg)\ and\ \bigg(3, 2 \dfrac{3}{4}\bigg)[/tex] . We can use these points to create two equations and then solve the system to find the a and k-values.
y = a(x - 2)² + k
[tex]\bigg(0, \dfrac{1}{2}\bigg)\\\rightarrow \dfrac{1}{2}=a(0-2)^2+k\\\\\rightarrow \dfrac{1}{2}=4a+k\\\\\rightarrow \dfrac{1}{2}-4a=k\\\\\\\bigg(3, 2\dfrac{3}{4}\bigg)\\\rightarrow 2\dfrac{3}{4}=a(3-2)^2+k\\\\\rightarrow 2\dfrac{3}{4}=a+k\\\\\rightarrow 2\dfrac{3}{4}-a=k\\\\\\\\\dfrac{1}{2}-4a=2\dfrac{3}{4}-a\\\\\dfrac{1}{2}-2\dfrac{3}{4}=4a-a\\\\\dfrac{2}{4}-\dfrac{11}{4}=3a\\\\-\dfrac{9}{4}=3a\\\\-\dfrac{9}{4\cdot 3}=\dfrac{3a}{3}\\\\ \bold{-\dfrac{3}{4}=a}[/tex]
Solve for k:
[tex]2\dfrac{3}{4}-a=k\\\\\\2\dfrac{3}{4}-\bigg(-\dfrac{3}{4}\bigg)=k\\\\\\\bold{3\dfrac{1}{2}=k}[/tex]
