XMocks · SAT Math · Quadratics

Read the graph from quadratic vertex form

Vertex form makes the turning point of a parabola visible. Once you identify the vertex and the sign of the leading coefficient, you can state the axis of symmetry and whether the graph has a maximum or a minimum before solving any equation.

Read each part of the expression

In f(x) = a(x − h)² + k, the vertex is (h, k) and the axis of symmetry is x = h. The square is smallest at zero, when x = h. If a is positive, the parabola opens upward and k is its minimum value. If a is negative, the parabola opens downward and k is its maximum value.

Pay attention to the sign inside the parentheses: (x + 3)² = (x − (−3))², so the x-coordinate is −3, not 3. The number outside the squared term gives the vertical position of the vertex.

Worked example: maximum and symmetric inputs

Let f(x) = −2(x + 3)² + 7.

  1. Rewrite the inside as x − (−3). The vertex is (−3, 7) and the axis is x = −3.
  2. The coefficient −2 is negative, so the graph opens downward. Its vertex is the highest point, making 7 the maximum value.
  3. To find where f(x) = 5, set −2(x + 3)² + 7 = 5. Then (x + 3)² = 1, so x = −4 or x = −2.

Check the two inputs

At x = −4 and x = −2, the squared quantity is 1. In each case f(x) = −2(1) + 7 = 5. The inputs lie one unit on either side of the axis x = −3, as symmetry predicts.

Common reading errors

  • Changing the sign of h incorrectly: the vertex’s x-coordinate is the opposite of the number written inside x − h.
  • Calling k a maximum automatically: k is a minimum when a > 0 and a maximum when a < 0.
  • Ignoring the scale factor: a changes how quickly the graph rises or falls, even though it does not move the vertex.

If an equation is given in standard form instead, completing the square can reveal vertex form. On a graph, the symmetry of the curve gives another way to check the axis and paired input values.

Practice: find the minimum and solve

For g(x) = 3(x − 2)² − 4, identify the vertex, state the minimum value, and solve g(x) = 8.

Show the worked answer

The vertex is (2, −4). Since the coefficient 3 is positive, the parabola opens upward and its minimum value is −4. Set 3(x − 2)² − 4 = 8, so (x − 2)² = 4. Thus x − 2 = 2 or x − 2 = −2, giving x = 4 or x = 0. Both are two units from the axis x = 2.

Practice and next steps

For a new vertex-form equation, record the vertex and opening direction first. Then use the equation or graph to answer the question asked.

This independently prepared XMocks lesson is not affiliated with or endorsed by the College Board.