SAT quadratics: three forms of a parabola and what each reveals

One parabola, three forms

A quadratic function makes a U-shaped graph called a parabola. The same parabola can be written three ways, and each form puts a different feature on display. The graph below shows one parabola, and the list after it gives three equivalent equations for it.

The study lab

Every form describes this parabola

y = x² − 2x − 8 with its vertex, x-intercepts, and y-intercept marked.

y = x² − 2x − 8

What to notice: The vertex sits halfway between the roots: (−2 + 4) ÷ 2 = 1. That symmetry is a fast way to find the vertex from factored form.

Standard form \(\displaystyle y = x^2 - 2x - 8\): the constant, −8, is the y-intercept.
Factored form \(\displaystyle y = (x - 4)(x + 2)\): the x-intercepts are 4 and −2.
Vertex form \(\displaystyle y = (x - 1)^2 - 9\): the vertex is (1, −9).

Play with vertex form

In vertex form, \(\displaystyle y = a(x - h)^2 + k\), the vertex is (h, k). Move the sliders to see how each number changes the graph. Watch the x-intercepts appear and disappear as the vertex crosses the x-axis.

Try it yourself · interactive

Parabola explorer

Vertex form

y = (x − 1)² − 4

Same parabola, standard form

y = x² − 2x − 3

a (stretch and direction)1h (horizontal shift)1k (vertical shift)-4

Vertex: (1, -4)
Axis of symmetry: x = 1
x-intercepts: -1, 3
Opens: Up ∪ (minimum)

Notice the sign flip: (x − 1)² puts the vertex at x = +1. A negative a turns the U upside down, so k becomes the maximum value instead of the minimum.

Which form answers the question?

SAT questions often ask for one feature and give you the “wrong” form. Recognize the target, then rewrite only as much as you need.

See the skill

Match the question to the form

Where does it cross the y-axis?

Standard form

Set x = 0: y = c. In y = x² − 2x − 8, the y-intercept is −8.

Where does it cross the x-axis?

Factored form

Set each factor to 0: x − 4 = 0 or x + 2 = 0.

What is the minimum or maximum?

Vertex form

k is the extreme value, reached when x = h.

Only standard form given?

x = −b ÷ (2a)

For x² − 2x − 8: x = 2 ÷ 2 = 1, then y = −9.

Use the visual as a reasoning aid: tie each choice to the relationship, grammar, or evidence the passage actually gives.

Count solutions with the discriminant

For \(\displaystyle ax^2 + bx + c = 0\), the discriminant is \(\displaystyle b^2 - 4ac\). It tells you how many times the parabola touches the x-axis without solving the equation.

For \(\displaystyle x^2 - 2x - 8 = 0\): \(\displaystyle (-2)^2 - 4(1)(-8) = 36\), which is positive, so there are two real solutions. For \(\displaystyle x^2 + 4x + 4 = 0\): \(\displaystyle 16 - 16 = 0\), so there is exactly one. For \(\displaystyle x^2 + 2x + 5 = 0\): \(\displaystyle 4 - 20 = -16\), so there are no real solutions.

Positive discriminant: two x-intercepts.
Zero: the vertex sits on the x-axis (one solution).
Negative: the parabola never reaches the x-axis (no real solutions).

Max and min questions in context

A ball’s height might be modeled by \(\displaystyle h(t) = -16(t - 2)^2 + 70\). Because a is negative, the parabola opens down, so the vertex is a maximum: the ball reaches 70 feet at t = 2 seconds. You do not need to expand anything.

If the model comes in standard form, either complete the square or use \(\displaystyle t = -b \div (2a)\) to locate the vertex, then substitute that value to find the height.

Your turn

Quick practice

The function f is defined by f(x) = −2(x − 3)² + 5. What is the maximum value of f?

Hint: The question asks for the value of the function, not the x-value where it occurs.

35−28

Practice: try these before checking the answers

Name the form you need before you do any algebra.

1. What are the x-intercepts of \(\displaystyle y = (x - 5)(x + 1)\)?
2. What is the vertex of \(\displaystyle y = x^2 + 6x + 2\)?
3. How many real solutions does \(\displaystyle 2x^2 - 3x + 4 = 0\) have?
4. What is the y-intercept of \(\displaystyle y = 3(x - 2)^2 - 7\)?

SAT quadratics: three forms of a parabola and what each reveals

What you'll take away