SAT Math – Advanced Math
Quadratic Graphs
Understanding parabolas through vertices, intercepts, and transformations
Quadratic graphs, called parabolas, are U-shaped curves representing functions of the form \(f(x) = ax^2 + bx + c\). On the SAT, you'll identify key features from equations and graphs—vertices, zeros, y-intercepts, axis of symmetry, domain, range, and transformations—connecting algebraic expressions to visual representations.
Success requires understanding how coefficients affect parabola shape and position, using vertex formulas to find maximum and minimum points, identifying x-intercepts as solutions, and interpreting transformations from different equation forms. These graphs aren't just mathematical curves—they model projectile paths, profit functions, satellite dishes, bridge arches, and any physical or economic relationship involving squared variables.
Understanding Parabola Features
Three Forms of Quadratic Functions
Each form reveals different information about the parabola.
Vertex form: \(f(x) = a(x - h)^2 + k\) → reveals vertex (h, k)
Factored form: \(f(x) = a(x - r_1)(x - r_2)\) → reveals zeros/x-intercepts
Coefficient a: Determines direction and width
Vertex - The Turning Point
The vertex is the highest or lowest point on the parabola.
From vertex form: Vertex is directly (h, k)
Maximum: If \(a < 0\) (opens down)
Minimum: If \(a > 0\) (opens up)
Axis of symmetry: Vertical line \(x = h\) through vertex
Intercepts
Points where parabola crosses axes.
X-intercepts (zeros): Solutions to \(ax^2 + bx + c = 0\)
Number of x-intercepts: 0, 1, or 2 depending on discriminant
Reading from graph: Count where parabola crosses x-axis
Domain and Range
Input and output values for parabola.
Range (opens up): \([k, \infty)\) where k is y-coordinate of vertex
Range (opens down): \((-\infty, k]\) where k is y-coordinate of vertex
Key insight: Range limited by vertex, domain unlimited
Essential Formulas and Properties
Finding the Vertex
From \(f(x) = ax^2 + bx + c\):
Vertex x-coordinate: \(h = -\frac{b}{2a}\)
Vertex y-coordinate: \(k = f\left(-\frac{b}{2a}\right)\)
From \(f(x) = a(x-h)^2 + k\): Vertex is (h, k) directly
Direction and Width
If \(a > 0\): Parabola opens upward (U-shape), vertex is minimum
If \(a < 0\): Parabola opens downward (∩-shape), vertex is maximum
If \(|a| > 1\): Parabola is narrower (more vertical)
If \(0 < |a| < 1\): Parabola is wider (more horizontal)
Axis of Symmetry
Equation: \(x = h\) (vertical line through vertex)
Divides parabola into mirror images
Midpoint between x-intercepts (if they exist)
Transformations
\(f(x) = a(x - h)^2 + k\) is \(f(x) = ax^2\) shifted:
• Right h units (if h > 0) or left |h| units (if h < 0)
• Up k units (if k > 0) or down |k| units (if k < 0)
• Reflected over x-axis if a is negative
Common Pitfalls & Expert Tips
❌ Sign confusion in vertex form
\(f(x) = (x + 3)^2\) has vertex at (-3, 0), not (3, 0). The form is \((x - h)^2\), so h = -3!
❌ Mixing up maximum and minimum
If parabola opens DOWN (a < 0), vertex is MAXIMUM. If opens UP (a > 0), vertex is MINIMUM. Check the sign of a!
❌ Forgetting y-intercept is the constant term
In \(f(x) = ax^2 + bx + c\), the y-intercept is (0, c). Just plug in x = 0 to find where it crosses y-axis!
❌ Incorrect range notation
Range includes the vertex! If vertex is (2, -3) and opens up, range is \([-3, \infty)\), not \((-3, \infty)\).
✓ Expert Tip: Use symmetry
Parabolas are symmetric about axis \(x = h\). If you know one point, you can find its mirror image across the axis!
✓ Expert Tip: Vertex form for transformations
Convert to vertex form to see transformations clearly. Completing the square reveals how the parabola has shifted.
✓ Expert Tip: Count x-intercepts from graph
Visually count where parabola crosses x-axis. Zero crossings = no real solutions, one = repeated root, two = distinct solutions.
Fully Worked SAT-Style Examples
What is the vertex of the parabola \(f(x) = x^2 - 6x + 13\)?
Solution:
Identify coefficients:
\(a = 1\), \(b = -6\), \(c = 13\)
Find x-coordinate of vertex:
\(x = -\frac{b}{2a} = -\frac{-6}{2(1)} = \frac{6}{2} = 3\)
Find y-coordinate:
\(f(3) = 3^2 - 6(3) + 13 = 9 - 18 + 13 = 4\)
Answer: Vertex is (3, 4)
For the function \(g(x) = -2(x + 1)^2 + 8\), identify the vertex, axis of symmetry, and whether it has a maximum or minimum.
Solution:
Recognize vertex form: \(a(x - h)^2 + k\)
Rewrite: \(-2(x - (-1))^2 + 8\)
\(h = -1\), \(k = 8\)
Vertex: (-1, 8)
Axis of symmetry: \(x = -1\)
Maximum or minimum:
\(a = -2 < 0\) → opens downward
Vertex is a MAXIMUM at y = 8
Answer: Vertex (-1, 8); axis x = -1; maximum value 8
What is the range of \(f(x) = 3(x - 2)^2 - 5\)?
Solution:
Find vertex:
Vertex form shows vertex at (2, -5)
Determine direction:
\(a = 3 > 0\) → opens upward
Vertex is minimum point
Determine range:
Minimum y-value: -5
Extends upward to infinity
Range: \([-5, \infty)\) or \(y \geq -5\)
Answer: \([-5, \infty)\)
What are the x-intercepts of \(h(x) = 2(x + 3)(x - 5)\)?
Solution:
Recognize factored form:
\(a(x - r_1)(x - r_2)\) where \(r_1\) and \(r_2\) are zeros
Set each factor equal to zero:
\(x + 3 = 0\) → \(x = -3\)
\(x - 5 = 0\) → \(x = 5\)
Interpretation:
X-intercepts are points where y = 0
Points: (-3, 0) and (5, 0)
Answer: x = -3 and x = 5 (or points (-3, 0) and (5, 0))
What is the y-intercept of \(f(x) = x^2 + 4x - 7\)?
Solution:
Y-intercept occurs when x = 0:
\(f(0) = 0^2 + 4(0) - 7 = -7\)
Shortcut:
In standard form \(ax^2 + bx + c\), y-intercept is always c
Here c = -7, so y-intercept is (0, -7)
Answer: (0, -7)
The graph of \(g(x) = (x - 4)^2 + 3\) is the graph of \(f(x) = x^2\) transformed in what way?
Solution:
Compare to vertex form: \(a(x - h)^2 + k\)
Here: \(h = 4\), \(k = 3\)
Horizontal shift:
\((x - 4)\) means shift RIGHT 4 units
Vertical shift:
\(+3\) means shift UP 3 units
Answer: Shifted right 4 units and up 3 units
A parabola has vertex at (3, 2) and passes through point (1, 6). What is another point on the parabola?
Solution:
Use symmetry property:
Axis of symmetry: \(x = 3\) (through vertex)
Find symmetric point:
Point (1, 6) is 2 units left of axis (3 - 1 = 2)
Mirror point is 2 units right of axis: 3 + 2 = 5
Same y-coordinate: 6
Answer: (5, 6)
Form Recognition Quick Guide
Standard Form
\(ax^2 + bx + c\)
Shows y-intercept (c)
Vertex Form
\(a(x-h)^2 + k\)
Shows vertex (h, k)
Factored Form
\(a(x-r_1)(x-r_2)\)
Shows x-intercepts
Quadratic Graphs: Visualizing Parabolic Relationships
Quadratic graphs transform abstract algebraic equations into visual parabolas that reveal maximum heights, minimum costs, optimal dimensions, and countless other insights impossible to see from equations alone. The SAT tests these graphical interpretations because they represent mathematical fluency essential for calculus, physics, engineering, and economics—understanding that the vertex of \(f(x) = -16t^2 + 64t + 80\) reveals when a projectile reaches maximum height, where a profit function achieves peak revenue, or when a quantity minimizes cost. The three forms—standard revealing y-intercepts instantly, vertex exposing turning points directly, factored displaying zeros immediately—each serve specific purposes, and skilled problem-solvers convert between them strategically. Recognizing that coefficient a controls both direction (positive opens up, negative down) and width (larger |a| creates narrower parabolas) connects algebraic parameters to visual shape. The axis of symmetry isn't just a line through the vertex but a powerful tool: knowing one point determines its mirror image, and understanding that x-intercepts are equidistant from this axis helps locate zeros when only one is given. Domain always spans all real numbers (parabolas extend infinitely left and right), but range is restricted by the vertex—extending from the turning point to infinity in the opening direction. Master these connections between equations and graphs: every factored form \(a(x-r_1)(x-r_2)\) produces a parabola crossing the x-axis at \(r_1\) and \(r_2\), every vertex form \(a(x-h)^2+k\) shifts the basic parabola h units horizontally and k units vertically, and every standard form \(ax^2+bx+c\) crosses the y-axis at c. These visual-algebraic connections transcend test preparation, equipping you to analyze any squared relationship graphically—from understanding why satellite dishes are parabolic (signals focus at vertex) to designing optimal bridge arches to modeling how businesses find production levels that maximize profit by locating the vertex of quadratic revenue functions.
