Quadratic Functions and Their Graphs Flashcards
What is a projectile?
An object propelled through air and then proceed with no additional force of their own.
How are paths of projectiles modeled?
By using quadratic functions.
What’s a quadratic function? What’s the standard form of a quadratic equation?
A function of degree 2 in one variable.
ax^2 + bx + c, where a is not equal to 0.
What’s the form of the graph of a quadratic function called?
A parabola.
What affects how a parabola opens?
The leading coefficient’s polarity.
If a < 0, the parabola opens downward, if a > 0 the parabola opens upward.
What is the vertex of a parabola?
The vertex, also called the turning point of a parabola, is the lowest point on the graph if it opens upward or the highest point on the graph if it opens downward.
What’s an axis of symmetry?
The vertical line, x = h, which passes through the vertex and divides the figure in half.
What is a parabola symmetric to?
The axis of symmetry.
What’s the vertex form of a quadratic function?
f(x) = a(x-h)^2 + k.
What do h and k mean in the vertex form of a quadratic function?
The vertex’s x and y coordinates.
What does h mean in the vertex form of a quadratic function?
The x-coordinate of the vertex.
What does k mean in the vertex form of a quadratic function?
The y-coordinate of the vertex.
Where does the axis of symmetry lie?
In the middle of the parabola, more specifically the h coordinate.
Describe, step by step, how the vertex form is derived.
- ax^2 + bx + c
- a(x^2 + bx/a) + c
- a(x^2 + bx/a + (b/2a)) + c - a(b/2a)^2
- a(x + b/2a)^2 + c - b^2/4a
- a(x - (-b/2a))^2 + c - b^2/4a
In the transformed version of ax^2 + bx + c, how are h and k represented?
h = -b/2a, k = c - b^2/4a
If the equation of a quadratic function is in standard form, how are the coordinates of the vertex found?
The x-coordinate of the vertex is found using -b/2a. The y-coordinate is obtained by solving the function at -b/2a, i.e f(-b/2a).
How is a quadratic function in standard form graphed? Let f be that function. Provide a step by step answer.
- Determine if the parabola opens up or down by checking the sign of a.
- Find the vertex using -b/2a for the x-coordinate and f(-b/2a) for the y-coordinate.
- Find the x-intercepts by setting the function equal to 0 and solving for x.
- Find the y-intercept by solving f(0).
- Plot the vertex, x-intercepts, and y-intercept. Connect the points with a smooth, continuous curve.
What are minimum/maximum values of quadratic equations?
The y-coordinate of the vertex gives the minimum or maximum value. If the leading coefficient is positive (graph opens upward), the vertex is a minimum. If the leading coefficient is negative (graph opens downward), the vertex is a maximum.
What are the applications of quadratic equations?
Problems involve finding the maximum or minimum value of a
quadratic function, as well as where this value occurs.
