chapter 8 Flashcards
What is the standard form of a quadratic function?
The standard form of a quadratic function is f(x) = ax^2 + bx + c.
True or False: The graph of a quadratic function is a straight line.
False.
What does the ‘a’ coefficient indicate in the quadratic function?
‘a’ indicates the direction of the parabola; if ‘a’ is positive, it opens upwards, and if ‘a’ is negative, it opens downwards.
Fill in the blank: The vertex of a quadratic function is the _____ point on its graph.
highest or lowest
What formula can be used to find the vertex of a quadratic function in standard form?
The vertex can be found using the formula x = -b/(2a).
What is the axis of symmetry for a quadratic function?
The axis of symmetry is a vertical line that passes through the vertex of the parabola, given by the equation x = -b/(2a).
True or False: The quadratic formula can be used to find the roots of a quadratic equation.
True.
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a).
What does the discriminant (b² - 4ac) tell us about the roots of a quadratic equation?
The discriminant indicates the nature of the roots: if positive, there are two real roots; if zero, one real root; if negative, no real roots.
In the context of a real-world scenario, what does a quadratic model often represent?
A quadratic model can represent projectile motion, profit/loss in business, or any situation where there is a maximum or minimum value.
Fill in the blank: The maximum or minimum value of a quadratic function occurs at the _____ of the parabola.
vertex
What is the vertex form of a quadratic function?
The vertex form is f(x) = a(x - h)² + k, where (h, k) is the vertex.
True or False: The y-intercept of a quadratic function can be found by evaluating f(0).
True.
What are the key features to identify when graphing a quadratic function?
Key features include the vertex, axis of symmetry, direction of opening, y-intercept, and x-intercepts (roots).
What is the significance of the roots of a quadratic function?
The roots are the values of x where the graph intersects the x-axis, representing solutions to the equation f(x) = 0.
Fill in the blank: The maximum height of a projectile can be modeled by a _____ function.
quadratic
What is the formula to convert from standard form to vertex form?
Complete the square to convert from standard form f(x) = ax^2 + bx + c to vertex form f(x) = a(x - h)² + k.
True or False: A quadratic function can have at most two real roots.
True.
What is a parabola?
A parabola is the U-shaped graph of a quadratic function.
How can you determine if a quadratic function opens upward or downward?
By examining the sign of the coefficient ‘a’: if ‘a’ > 0, it opens upward; if ‘a’ < 0, it opens downward.
Fill in the blank: The point where the parabola intersects the y-axis is called the _____.
y-intercept
What are the steps to graph a quadratic function?
Identify the vertex, axis of symmetry, y-intercept, and roots; then plot these points and draw a smooth curve.
True or False: The vertex of a parabola is always located at the origin.
False.
What are the key characteristics of a quadratic function?
Key characteristics include direction of opening, vertex, axis of symmetry, y-intercept, and roots.
How do you find the x-intercepts of a quadratic function?
Set f(x) = 0 and solve the quadratic equation using factoring, completing the square, or the quadratic formula.
Fill in the blank: A quadratic function can model the path of a _____.
projectile
What is the vertex of the quadratic function f(x) = 2(x - 3)² + 5?
(3, 5).
True or False: The vertex form of a quadratic function makes it easy to identify the vertex.
True.
What does it mean for a quadratic equation to be in factored form?
Factored form is expressed as f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots.
What is the relationship between the roots and the vertex of a quadratic function?
The vertex lies on the axis of symmetry, which is equidistant from the roots.
Fill in the blank: The maximum value of a downward-opening parabola occurs at the _____.
vertex
How can real-world data often be modeled using quadratic functions?
Real-world data with a parabolic trend, such as profit maximization or projectile motion, can often be modeled using quadratic functions.
What is the effect of changing the ‘b’ value in the quadratic equation?
Changing ‘b’ affects the position of the vertex and the axis of symmetry, but not the direction of opening.
True or False: You can have a quadratic function without real roots.
True.
What is the relationship between the vertex and the discriminant?
The vertex provides the maximum or minimum value, while the discriminant indicates the number and type of roots.
