4.2 Quadratic Graphs & Modeling
Key Takeaways
- The axis of symmetry of y = ax^2 + bx + c is the vertical line x = -b/(2a), and the vertex lies on it.
- Vertex form y = a(x - h)^2 + k shows the vertex (h, k) directly; the parabola opens up when a is positive (minimum) and down when a is negative (maximum).
- Factored form y = a(x - r)(x - s) shows the zeros r and s directly; the y-intercept is the value of y when x = 0.
- Average rate of change over [x1, x2] equals (f(x2) - f(x1)) / (x2 - x1), the slope of the secant line.
- In projectile models like h(t) = -16t^2 + bt + c, the vertex gives maximum height and a zero gives when the object lands.
Anatomy of a Parabola
Every quadratic graphs as a parabola with a single turning point called the vertex. The vertical line through the vertex is the axis of symmetry; the parabola is a mirror image across it.
- Opens up when a is positive: the vertex is a minimum.
- Opens down when a is negative: the vertex is a maximum.
- Zeros (roots, x-intercepts) are where y = 0.
- y-intercept is the y-value when x = 0, which equals c in standard form.
Finding the Vertex from Standard Form
For y = ax^2 + bx + c, the axis of symmetry is the line:
x = -b/(2a)
The vertex's x-coordinate is that value; substitute it back to get the y-coordinate.
Example: y = x^2 - 6x + 5. Here a = 1, b = -6, so x = -(-6)/(2 times 1) = 6/2 = 3. Substitute: y = (3)^2 - 6(3) + 5 = 9 - 18 + 5 = -4. The vertex is (3, -4), a minimum because a is positive.
What is the axis of symmetry of y = 2x^2 + 8x - 1?
Vertex Form and Transformations
Vertex form is y = a(x - h)^2 + k, where the vertex is (h, k). This form makes shifts easy to read.
For y = (x + 1)^2 - 3, rewrite x + 1 as x - (-1), so h = -1 and k = -3. The vertex is (-1, -3). Compared with the parent y = x^2, the graph shifts 1 unit left and 3 units down.
Transformation summary for y = a(x - h)^2 + k versus y = x^2:
- h shifts horizontally: positive h moves right, negative h moves left.
- k shifts vertically: positive k moves up, negative k moves down.
- a stretches the graph and, when negative, reflects it over its axis.
Comparing the Three Forms
Each algebraic form of a quadratic reveals a different feature instantly. Choosing the right form is a frequent Regents skill.
| Form | Equation | Reveals directly |
|---|---|---|
| Standard | y = ax^2 + bx + c | y-intercept c; use x = -b/(2a) for vertex |
| Vertex | y = a(x - h)^2 + k | Vertex (h, k); max or min value |
| Factored | y = a(x - r)(x - s) | Zeros x = r and x = s |
So if a question asks for the zeros, factored form is best; if it asks for the maximum height, vertex form is best.
Which form of a quadratic most directly shows that its zeros are 4 and -1?
Interpreting Quadratic Models
Many Regents word problems are projectile motion: h(t) = -16t^2 + v*t + h0, where -16 reflects gravity in feet per second squared, v is initial upward velocity, and h0 is starting height.
Example: h(t) = -16t^2 + 64t + 5. The maximum height occurs at the vertex. Axis: t = -64/(2 times -16) = -64/-32 = 2 seconds. Height: h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = 69 feet. So the ball reaches a maximum of 69 feet at t = 2 s.
Domain, Zeros, and Area Models
In the projectile above, the object lands when h(t) = 0. A reasonable domain runs from t = 0 (launch) to the positive zero (impact), because negative time and negative height have no physical meaning.
Area models are also quadratic. A rectangle with length x + 4 and width x - 2 has area (x + 4)(x - 2) = x^2 + 2x - 8. If the area is 48, solve x^2 + 2x - 8 = 48, so x^2 + 2x - 56 = 0, factor (x + 8)(x - 7) = 0. Since a length needs x greater than 2, x = 7.
Average Rate of Change
Unlike a line, a parabola's steepness changes. The average rate of change of a function f over an interval from x1 to x2 is:
(f(x2) - f(x1)) / (x2 - x1)
This is the slope of the secant line joining the two points. For f(x) = x^2 over the interval from x = 1 to x = 3: f(3) = 9, f(1) = 1, so the rate is (9 - 1)/(3 - 1) = 8/2 = 4. Over from x = 0 to x = 2 it would be (4 - 0)/(2 - 0) = 2, showing the rate grows as x increases.
Reading Graphs in Context
Regents items reward interpretation. When a parabola models height versus time, the vertex is the highest point, a zero is when the object is at ground level, and the y-intercept is the starting height. When a < 0 the model has a maximum; when a > 0 it has a minimum.
For y = -2(x - 4)^2 + 6, a = -2 is negative, so the parabola opens down and the vertex (4, 6) is the maximum; the largest output is 6, reached at x = 4. Always tie the math feature back to what the variable means in the problem.
A Regents trap is treating a maximum-height question as if it asked for time, or reporting only the t-value when the question wants the height. Read the vertex coordinates carefully: the x-coordinate answers "when" and the y-coordinate answers "how high."