Parabolas and quadratic graphs — Digital SAT Math practice
Vertex, axis of symmetry and the three quadratic forms.
Key terms
- Vertex: The highest or lowest point on a parabola where the graph changes direction.
- Axis of Symmetry: The vertical line passing through the vertex that divides the parabola into two mirror images.
- Standard Form: The polynomial expression y = ax^2 + bx + c used to easily identify coefficients and the y-intercept.
- Leading Coefficient: The value a in quadratic equations that determines the width and direction of the parabola.
Common traps
- Forgetting to flip the sign of h when reading the vertex from y = a(x - h)^2 + k.
- Confusing the axis of symmetry formula with the quadratic formula roots.
- Assuming the y-intercept is always zero instead of reading it directly from c in standard form.
- Misidentifying the direction by looking at the sign of b instead of a.
Key ideas to know
Finding the Vertex from Vertex Form
Given the quadratic equation y = 3(x + 4)2 - 5, what are the coordinates of the vertex?
The vertex is (h, k). Since the form is y = a(x - h)2 + k, rewrite x + 4 as x - (-4) to get h = -4. Thus, the vertex is (-4, -5).
Watch out: Misreading the sign of h; the x-coordinate is always the opposite of what appears inside the parentheses.
Axis and Intercept in Standard Form
For the quadratic function y = 2x2 - 8x + 3, what is the equation of the axis of symmetry and the y-intercept?
The axis of symmetry is x = -b/(2a) = -(-8)/(2*2) = 2. The y-intercept is c, which equals 3. So the axis is x = 2 and the y-intercept is (0, 3).
Watch out: Confusing the axis of symmetry with the vertex's y-coordinate or forgetting that c represents the y-intercept directly.
Using Factored Form to Find the Axis
A quadratic has x-intercepts at 1 and 9. Which quadratic form is most efficient for finding the equation of its axis of symmetry, and what is that line?
The factored form y = a(x - p)(x - q) shows intercepts p and q. The axis of symmetry lies exactly halfway between them: x = (1 + 9)/2 = 5. Use the factored form to quickly average the roots.
Watch out: Assuming the axis of symmetry equals one of the intercepts or averaging them incorrectly when signs differ.
Minimum or Maximum Based on a
A student claims that the quadratic y = -5x2 + 10x + 3 has a minimum value at its vertex because it is a parabola. What is wrong with this reasoning?
The statement is wrong because the leading coefficient a = -5 is negative, so the parabola opens downward and has a maximum, not a minimum. A minimum only occurs when a > 0.
Watch out: Forgetting that the sign of a determines direction; assuming all parabolas have minima or mixing up the conditions for min vs max.
Sample questions
Question 1
What is the maximum value of the function f(x) = −1x² − 12x − 29 ?
- −29
- −7
- 7
- −6
Show answer and explanation
Answer: 7
Vertex x = −b/(2a) = −6; f(−6) = 7. The maximum value is 7.