GSE Algebra 1 - Unit 3 Quadratic Functions

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What are the roots, zeros, x-intercepts, or solutions of a quadratic function?

Where the parabola crosses the x-axis.

What is a perfect square trinomial?

A trinomial expression where the 'a' and 'c' values are perfect squares and 'b' is twice the square root of 'a' and 'c'.

What does the 'c' value in the standard form of a quadratic represent?

The y-intercept.

What do 'r' and 's' represent in the factored form of a quadratic?

They are the zeros of the function. Signup and view all the answers

What is the vertex of a parabola?

The highest or lowest point in a parabola. Signup and view all the answers

What is the quadratic formula used for?

To find the zeros of a quadratic equation. Signup and view all the answers

What is a parabola?

The curved graph of a quadratic function. Signup and view all the answers

What do (h, k) represent in the vertex form of a parabola?

The vertex of the function. Signup and view all the answers

What are vertical shifts of a parabola?

Translating a parabola up or down. Signup and view all the answers

What are horizontal shifts of a parabola?

Translating a parabola left or right. Signup and view all the answers

What is the difference of squares?

A way of factoring requiring 'a' and 'c' values being perfect squares and a subtraction of the 'c' value. Signup and view all the answers

What does the vertex of a parabola indicate?

The highest or lowest point of a parabola. Signup and view all the answers

What are reflections of a parabola?

A 'flip' of a parabola over the x-axis. Signup and view all the answers

What does dilation of a parabola refer to?

A vertical 'stretch' or 'shrink' of the parabola. Signup and view all the answers

What is the axis of symmetry in relation to a parabola?

The line that cuts the parabola into two symmetrical halves. Signup and view all the answers

What is the discriminant in quadratic equations?

A method of determining how many zeros a quadratic equation has prior to solving. Signup and view all the answers

What is the y-intercept of a parabola?

Where the parabola crosses the y-axis. Signup and view all the answers

What is factoring in the context of quadratics?

The process of writing a standard form equation into factored form. Signup and view all the answers

Study Notes

Roots/Zeros/X-Intercepts/Solutions

Points where a parabola intersects the x-axis, indicating the solutions of the quadratic equation.
A quadratic function can have zero, one, or two x-intercepts.

Perfect Square Trinomial

A trinomial where both "a" and "c" values are perfect squares.
The "b" value is calculated as twice the product of the square roots of "a" and "c".

Standard Form of a Quadratic

Represented as ( ax^2 + bx + c ).
The "c" value corresponds to the y-intercept of the quadratic function.

Factored Form of a Quadratic

Written as ( a(x - r)(x - s) ), with "r" and "s" representing the zeros of the quadratic function.

Vertex of a Parabola

The vertex is the peak (maximum) or trough (minimum) of the parabola.
It marks the point of inflection where the curve changes direction.

Quadratic Formula

A pivotal formula given by ( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} ) used to determine the zeros of a quadratic equation.

Parabola

The graphical representation of a quadratic function, characterized by its U-shaped curve.

Vertex Form of a Parabola

Expressed as ( y = a(x - h)^2 + k ), where (h, k) is the vertex.
Vertex form reveals transformations applied to the basic parabola.

Vertical Shifts of a Parabola

Movement up or down the y-axis.
Represented as ( f(x) = x^2 + a ) (upward) or ( f(x) = x^2 - a ) (downward).

Horizontal Shifts of a Parabola

Movement along the x-axis.
Represented as ( f(x) = (x + 4)^2 ) (left shift) or ( f(x) = (x - 4)^2 ) (right shift).

Difference of Squares

A factoring method applicable when both "a" and "c" are perfect squares and their relationship involves subtraction.
It is expressed as ( a^2 - b^2 = (a + b)(a - b) ).

Minimum or Maximum

Referred to as the vertex; points can be either the lowest (minimum) or the highest (maximum) in a quadratic function.

Reflections of a Parabola

Flipping the parabola over the x-axis changes its orientation.
Represented by ( f(x) = -x^2 ).

Dilation of a Parabola

Refers to stretching or compressing the parabola vertically.
Increases or decreases the width of the parabola based on the coefficient ( a ) in ( f(x) = ax^2 ) or ( f(x) = \frac{1}{a}x^2 ).

Axis of Symmetry

The vertical line (( x = h )) that divides the parabola into two equal parts through the vertex.

Discriminant

A formula ( b^2 - 4ac ) that indicates the number of real zeros (solutions) in a quadratic equation before solving it.

Y-intercept of a Parabola

The point where the parabola intersects the y-axis, corresponding to the value of ( c ) in the standard form.

Factoring

The method of re-expressing a quadratic equation from its standard form to its factored form.
Applicable for different values of "a", including when ( a = 1 ) and special cases.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge on modeling and analyzing quadratic functions with our flashcards. This quiz covers essential terms such as roots, perfect square trinomials, and standard form of quadratic equations.