Quadratic Functions and Their Graphs

Question 1

What is a projectile?

Answer 1

An object propelled through air and then proceed with no additional force of their own.

Question 2

How are paths of projectiles modeled?

Answer 2

By using quadratic functions.

Question 3

What’s a quadratic function? What’s the standard form of a quadratic equation?

Answer 3

A function in one variable whose highest exponent is 2.
ax^2 + bx + c, where a is not equal to 0.

Question 4

What’s the form of the graph of a quadratic function called?

Answer 4

A parabola.

Question 5

What affects how a parabola opens?

Answer 5

The leading coefficient’s polarity.
If a < 0, the parabola opens downward, if a > 0 the parabola opens upward.

Question 6

What is the vertex of a parabola?

Answer 6

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.

Question 7

What’s an axis of symmetry?

Answer 7

The vertical line, x = h, which passes through the vertex and divides the figure in half.

Question 8

What is a parabola symmetric to?

Answer 8

The axis of symmetry.

Question 9

What’s the vertex form of a quadratic function?

Answer 9

f(x) = a(x-h)^2 + k.

Question 10

What do h and k mean in the vertex form of a quadratic function?

Answer 10

The vertex’s x and y coordinates.

Question 11

What does h mean in the vertex form of a quadratic function?

Answer 11

The x-coordinate of the vertex.

Question 12

What does k mean in the vertex form of a quadratic function?

Answer 12

The y-coordinate of the vertex.

Question 13

Where does the axis of symmetry lie?

Answer 13

In the middle of the parabola, more specifically the h coordinate.

Question 14

Describe, step by step, how the vertex form is derived.

Answer 14

1. ax^2 + bx + c
2. a(x^2 + bx/a) + c
3. a(x^2 + bx/a + (b/2a)) + c - a(b/2a)^2
4. a(x + b/2a)^2 + c - b^2/4a
5. a(x - (-b/2a))^2 + c - b^2/4a

Question 15

In the transformed version of ax^2 + bx + c, how are h and k represented?

Answer 15

h = -b/2a, k = c - b^2/4a

Question 16

If the equation of a quadratic function is in standard form, how are the coordinates of the vertex found?

Answer 16

The x-coordinate of the vertex is found using -b/2a. The y-coordinate is obtained by substituting that x-value into the function.

Question 17

How is a quadratic function in standard form graphed? Let f be that function. Provide a step by step answer.

Answer 17

1. Determine if the parabola opens up or down by checking the sign of a.
2. Find the vertex using -b/2a for the x-coordinate and f(-b/2a) for the y-coordinate.
3. Find the x-intercepts by setting the function equal to 0 and solving for x.
4. Find the y-intercept by solving f(0).
5. Plot the vertex, x-intercepts, and y-intercept. Connect the points with a smooth, continuous curve.

Question 18

What are minimum/maximum values of quadratic equations?

Answer 18

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.

Question 19

What are the applications of quadratic equations?

Answer 19

Problems involve finding the maximum or minimum value of a
quadratic function, as well as where this value occurs.