Coordinate Geometry and Conics · Lesson 4

Parabolas: vertex form, axis and direction

Vertex form. In y=a(x−h)2+ky = a(x - h)^2 + k the vertex is (h,k)(h, k) and the axis of symmetry is the vertical line x=hx = h. If a>0a > 0 the parabola opens up and the vertex is its lowest point; if a<0a < 0 it opens down and the vertex is its highest point. As with circles, the sign in the parentheses is opposite to h: (x+3)2(x + 3)^2 means h=−3h = -3.

Standard form. For y=ax2+bx+cy = ax^2 + bx + c, the axis is x=−b2ax = -\frac{b}{2a}. Substitute that x back in to get the y-coordinate of the vertex, which is the minimum or maximum value of y.

Sideways parabolas. When y is the squared variable, x=a(y−k)2+hx = a(y - k)^2 + h, the vertex is (h,k)(h, k) and the axis is the horizontal line y=ky = k. It opens right if a>0a > 0 and left if a<0a < 0. Check which variable is squared before you decide on up, down, left or right.

Worked example Easy

Which statement describes the graph of y=−2(x+3)2+5y = -2(x + 3)^2 + 5 in the standard (x, y) coordinate plane?

  1. Vertex (3,5)(3, 5), opens up
  2. Vertex (3,5)(3, 5), opens down
  3. Vertex (−3,5)(-3, 5), opens up
  4. Vertex (−3,5)(-3, 5), opens downAnswer

How to solve it

  1. Match y=a(x−h)2+ky = a(x - h)^2 + k: x+3=x−(−3)x + 3 = x - (-3), so h=−3h = -3, and k=5k = 5.
  2. The vertex is (−3,5)(-3, 5).
  3. a=−2<0a = -2 < 0, so the parabola opens down.

Why each choice is right or wrong

  • A. Incorrect. Vertex (3,5)(3, 5), opens up makes two slips: it flips the sign of h and ignores the negative value of a.
  • B. Incorrect. Vertex (3,5)(3, 5) flips the sign of h. The square (x+3)2(x + 3)^2 means h=−3h = -3, although the direction, down, is right.
  • C. Incorrect. The vertex (−3,5)(-3, 5) is right, but opens up ignores the sign of a=−2a = -2. A negative a makes the parabola open down.
  • D. Correct. In vertex form, h=−3h = -3 and k=5k = 5, so the vertex is (−3,5)(-3, 5), and a=−2a = -2 is negative, so the parabola opens down.

Worked example Medium

What is the minimum value of y=2x2−12x+7y = 2x^2 - 12x + 7?

  1. −20-20
  2. −11-11Answer
  3. 33
  4. 77

How to solve it

  1. Axis: x=−b2a=−−122(2)=3x = -\frac{b}{2a} = -\frac{-12}{2(2)} = 3.
  2. Substitute: y=2(3)2−12(3)+7=18−36+7=−11y = 2(3)^2 - 12(3) + 7 = 18 - 36 + 7 = -11.
  3. a=2>0a = 2 > 0, so the parabola opens up and −11-11 is the minimum.

Why each choice is right or wrong

  • A. Incorrect. −20-20 squares 3 but drops the coefficient 2: 9−36+79 - 36 + 7. The first term is 2(3)2=182(3)^2 = 18.
  • B. Correct. The vertex is at x=−−124=3x = -\frac{-12}{4} = 3, and y=18−36+7=−11y = 18 - 36 + 7 = -11. Since a>0a > 0, that is the lowest value of y.
  • C. Incorrect. 3 is the x-coordinate of the vertex, where the minimum happens. The question asks for the minimum value of y.
  • D. Incorrect. 7 is the value of y when x=0x = 0, the y-intercept. The graph keeps falling to the right of x=0x = 0 until x=3x = 3.

Trap Which variable is squared

y=(x−1)2y = (x - 1)^2 opens up, but x=(y−1)2x = (y - 1)^2 opens right, and its vertex lists the constant term first: x=−(y−1)2+3x = -(y - 1)^2 + 3 has vertex (3,1)(3, 1), not (1,3)(1, 3). Also watch the vertex sign: (x+3)2(x + 3)^2 means h=−3h = -3.

Calculator When a calculator is faster

Graphing is a quick check. In the built-in Desmos calculator (if you test online) you can type y=2x^2-12x+7 and click the curve to see its lowest point, and you can even type a sideways parabola such as x=-(y-1)^2+3 as it is. A handheld graphing calculator that graphs only y = … equations needs y in terms of x, so sideways parabolas are faster by hand there.