# 9.4 – Parabolas

## Key Terms

• Standard Form – An equation of a parabola in the forms:
• $y=ax^{a}+bx+c$ (opens up or down)
• $x=ay^{a}+by+c$ (opens left or right)
• Vertex Form – An equation of a parabola in the form:
• $y=a(x-h)^{2}+v$ (opens up or down)
• $x=a(y-v)^{2}+h$ (opens left or right)
• h = horizontal
• v = vertical

## Notes

Parabolas
• Four kinds of parabolas
• Opening left
• Opening right
• Opening up
• Opening down
Vertex of a Parabola
• The vertex of a parabola is the point on the parabola where the curve makes the sharpest turn.
• It’s where the curve changes directions.
Coefficients
• If the coefficient is close to zero, then the parabola is wide and shallow.
• If the coefficient is far away from zero (a large positive number or a negative number much less than zero), then the parabola is steep and narrow.
• The coefficient can’t be zero. If it’s zero, then it’s not a parabola. It’s a line.
• The closer the coefficient gets to zero, the closer the parabola gets to being a line.

Parabolas that Open Up or Down (Centered at the Origin)  Parabolas that Open Left or Right (Centered at the Origin)
• Coefficient of $x^{2}$
• If the coefficient of $x^{2}$ is positive, then the parabola opens upward.
• If the coefficient of $x^{2}$ is negative, then the parabola opens downward.
• Coefficient of $y^{2}$
• When the coefficient of $y^{2}$ is positive, the parabola opens to the right.
• When the coefficient of $y^{2}$ is negative, the parabola opens to the left.
• Equation: $y=ax^{2}$
• (0, 0) = Vertex of the parabola
• (x, y) = Any point on the parabola
• a = A nonzero number
• Equation: $x=ay^{2}$
• (0, 0) = Vertex of the parabola
• (x, y) = Any point on the parabola
• a = A nonzero number

Parabolas Not Centered at the Origin
• Vertex – for parabolas, (h, v) is the vertex because there is no center (like in a circle).
• How to Shift a Parabola’s Vertex from the Origin (0, 0) to (h, v)
1. Subtract h from the x-term and v from the y-term.
2. Isolate the variable that is not squared on one side of the equation.
Parabolas that Open Up or Down (Not Centered at the Origin)  Parabolas that Open Left or Right (Not Centered at the Origin)
• Equation: $y=a(x-h)^{2}+v$
• Equation: $x=a(y-v)^{2}+h$
• Coefficient
• Find the y-value of the point 1 unit to the right of the vertex.
• Subtract the y-value of the point at the vertex from the value you found in step 1.
• Vertex: (h, v)
• 1 Unit to the Right: (h+1, v_1)
• Finding a: new y-value minus original y-value: $a=v_1-v$
• Coefficient
• Find the x-value of the point 1 unit above the vertex of the parabola.
• Subtract the x-value of the point at the vertex from the value you found in step 1.
• Vertex: (h, v)
• 1 Unit Above: (h_1, v+1)
• Finding a: new x-value minus original x-value: $a=h_1-h$

Convert from Vertex to Standard Form  Convert from Standard Form to Vertex Form
1. Begin with the equation for a parabola
2. Simplify
3. Expand
4. Simplify

• Example 1
1. $y=4(x-2)^{2}+(-1)$
2. $y=4(x-2)^{2}-1$
3. $y=4(x^{2}-4x+4)-1$
4. $4x^{2}-16x+15$

• Example 2
1. $x=3(y-(-1))^{2}+2$
2. $x=3(y+1)^{2}+2$
3. $x=3(y^{2}+2y+1)+2$
4. $x=3y^{2}+6y+5$
1. Begin with the equation for a parabola
2. Subtract the constant on both sides
3. Divide both sides by the coefficient of the $x^{2}$ term
4. Complete the square of the right side & add it to both sides
5. Factor the right side into 2 identical binomials and write them as $(x-h)^{2}$ or $(y-v)^{2}$
6. Is there a constant term on the left side of the equation (separate from the fraction)? If so, take the inverse operation of it on both sides
7. Is there a denominator on the left side of the equation? If so, multiply it on both sides
8. Is there another constant term on the left side of the equation? If so, take the inverse operation of it on both sides.

• Example
1. $y=2x^{2}+4x+3$
2. $y-3=2x^{2}+4x$
3. $\frac{y-3}{2}=x^{2}+2x$
4. $\frac{y-3}{2}+1=x^{2}+2x+1$
5. $\frac{y-3}{2}+1=(x+1)^{2}$
6. $\frac{y-3}{2}=(x+1)^{2}-1$
7. $y-3=2(x+1)^{2}-2$
8. $y=2(x+1)^{2}+1$