Direction, vertex, roots — the picture hands you every number the equation is hiding. Then one marked point pins down the rest.
Read the graph
What is the vertex of the graphed parabola?
The vertex is the turning point — here the lowest point, since the curve opens upward: (2, −1). It sits one square below the x-axis, so the y must be negative — that kills C. D picks a root and the y-intercept, two different features mashed together.
a > 0
the vertex is the lowest point
a < 0
the vertex is the highest point
In a which-equation-matches question, direction alone usually kills two of the four choices — before you read a single coordinate.
Read both coordinates, signs included.
vertex = (2, −1)
the lowest point of the curve
axis of symmetry: x = 2
the dashed vertical through the vertex
Everything on the curve mirrors across that dashed line. If you know a point one square left of the axis, you also know the point one square right of it — the graph gives you every second point free.
roots: x = 1 and x = 3
the two crossing points
(1 + 3) ÷ 2 = 2
the axis sits midway between them ✓
Roots and axis must agree. If the midpoint of your roots is not your axis, one of the three numbers was misread.
y = a(x − h)² + k, where (h, k) is the vertex. The marked point does the rest.
y = a(x − 2)² − 1
vertex (2, −1) dropped in
3 = a(0 − 2)² − 1
using the marked point (0, 3)
3 = 4a − 1 → a = 1
solve the one unknown
y = (x − 2)² − 1
the graphed equation
The vertex is at x = 2, so the hand writes a “+ 2”: y = (x + 2)² − 1. Test it — that bracket is zero when x = −2, so its vertex is (−2, −1), the mirror image of the graph. The bracket is (x − h): a vertex to the right of the y-axis shows a minus sign. The k outside keeps its own sign, so − 1 stays − 1.
Vertex position + direction tells you how many times it crosses — the discriminant, read visually.
2 roots
it must climb through the axis twice
1 root
it touches and turns back
0 roots
it floats — never reaches the axis
A level line y = k works the same way. Slide the axis up or down to height k and count the crossings. For our curve (opens up, lowest point y = −1): y = k meets it twice when k > −1, once when k = −1 (at the vertex), and never when k < −1. For a curve that opens down, the inequalities flip: two points when k is below the highest value.
A wide, downward curve. Opens down, vertex (−1, 3), passes through (1, 1).
y = a(x + 1)² + 3
vertex form, h = −1 gives (x + 1)
1 = a(1 + 1)² + 3 = 4a + 3
the marked point
4a = −2 → a = −12
negative, as it opens down ✓
y = −½(x + 1)² + 3
the graphed equation
What is different: a need not be 1 or −1. A curve wider than y = x² gives a fraction, and the sign must match the direction.
One root that only touches. The parabola touches the x-axis at x = −1 and crosses the y-axis at 2.
vertex (−1, 0)
a touch point is the vertex
y = a(x + 1)²
k = 0
2 = a(0 + 1)² → a = 2
the y-intercept (0, 2)
y = 2(x + 1)² = 2x² + 4x + 2
expand if the choices need it
What is different: the single root counts twice — factored form is 2(x + 1)(x + 1) — and vertex form and factored form become the same thing.
Roots off the grid. Opens up, vertex (2, −3), passes through (0, 1). Find the roots.
1 = a(0 − 2)² − 3 → a = 1
the marked point
0 = (x − 2)² − 3 → (x − 2)² = 3
roots: set y = 0
x − 2 = ±√3
two signs
x = 2 + √3 ≈ 3.7 or 2 − √3 ≈ 0.3
symmetric about x = 2 ✓
What is different: the crossings fall between gridlines, so we read the vertex and compute the roots rather than guess them.
Guessing roots of 4 and 0 fails the test: (4 − 2)² − 3 = 1, not 0. The two roots must still average to the axis, x = 2.
A graphed parabola crosses the x-axis at −3 and 1 and the y-axis at −6. Find its equation and its lowest value.
y = a(x + 3)(x − 1)
root r gives the factor (x − r)
−6 = a(3)(−1) = −3a → a = 2
the y-intercept pins down a
y = 2(x² + 2x − 3) = 2x² + 4x − 6
expand if the choices are in standard form
axis x = (−3 + 1) ÷ 2 = −1; y = 2(2)(−2) = −8
the minimum sits on the axis
(x + 3)(x − 1) has the right roots but crosses the y-axis at −3, not −6. Always test a third point; a marked point that is not the vertex is there to give you a, not the minimum.
Signs of a, b and c straight from the picture, for y = ax² + bx + c:
| Look at | It tells you | Our curve |
|---|---|---|
| opens up or down | sign of a | up, so a > 0 |
| the y-intercept | sign of c | −6, so c < 0 |
| vertex left or right of the y-axis | x = −b2a: left means a and b have the same sign; right means opposite signs | left (x = −1), so b > 0 |
Parabola-graph items ask for the matching equation, a value off the grid (y-intercept, minimum), how many times a line y = k meets it, or the signs of a, b, c. Use vertex form when the vertex is clear and factored form when the roots are; then check one more marked point before you commit.
Read the graph
Which equation matches the graph?
It opens downward, so a is negative — A and C die on direction alone. The vertex is (−1, 4), and h = −1 goes in as (x + 1), giving y = −(x + 1)² + 4. B parks the vertex at x = +1 — the sign of h flips on the way into the bracket. Check: at x = 0, −(1) + 4 = 3, exactly the marked y-intercept ✓.
