In this case, the equation is in vertex form. To graph the parabola, we need to determine the y-intercept, x-intercept(s), and the vertex.
Vertex of Parabola:
Vertex form: y = a(x + h)² - k
- ⇒ [y = 3(x + 2)² - 1] and [y = a(x - h)² - k]
- ⇒ Vertex: (h, k) ⇒ (-2, -1)
X-intercept(s) of Parabola:
Assume "y" as 0.
[tex]\implies 0 = 3(x + 2)^{2} - 1[/tex]
[tex]\implies 1 = 3(x + 2)^{2}[/tex]
Take square root both sides and simplify:
[tex]\implies\sqrt{1} = \sqrt{3(x + 2)^{2}}[/tex]
[tex]\implies1 = \sqrt{3 \times (x + 2) \times (x + 2)}[/tex]
[tex]\implies1 = (x + 2) \times \sqrt{3[/tex]
Divide √3 both sides:
[tex]\implies \±\huge\text{(}\dfrac{1 }{\sqrt{3}} \huge\text{)} = (x + 2)[/tex]
[tex]\implies \±\huge\text{(}\dfrac{\sqrt{1} }{\sqrt{3}} \huge\text{)} = (x + 2)[/tex]
Multiply √3 to the numerator and the denominator:
[tex]\implies \±\huge\text{(}\dfrac{\sqrt{1} \times \sqrt{3} }{\sqrt{3\times \sqrt{3}}} \huge\text{)} = (x + 2)[/tex]
[tex]\implies \±\huge\text{(}\dfrac{\sqrt{3} }{3}} \huge\text{)} = (x + 2)[/tex]
[tex]\implies \±\huge\text{(}\dfrac{\sqrt{3} }{3}} \huge\text{)} - 2 =x[/tex]
[tex]\implies x = \underline{-2.6...} \ \text{and} \ \underline{-1.4...} \ \ \ \ (\text{Nearest tenth})[/tex]
Y-intercept of Parabola:
Assume "x" as 0.
- ⇒ y = 3(0 + 2)² - 1
- ⇒ y = 3(2)² - 1
- ⇒ y = 3(4) - 1
- ⇒ y = 12 - 1
- ⇒ y = 11
y-intercept = 11 ⇒ (0, 11)
Determining the direction of the parabola:
Since the first cooeficient is positive (+3), the direction of the parabola will be upwards.
Determining the axis of symmetry line:
Axis of symmetry line: x-coordinate of vertex
Axis of symmetry line: x = -2
Plot the following on the graph:
- y-intercept ---> 11 -----> (0, 11)
- x-intercept(s) -----> -1.4 and -2.6 -------> (-1.4, 0) and (-2.6, 0)
- Vertex -------> (-2, -1)
- Axis of symmetry ----> x = -2
Refer to graph attached.
