Quadratics and Quadratic Transformations

Front

What is the vertex of a quadratic function?

Back

The vertex is the highest or lowest point of the parabola, depending on its orientation. It can be found using the formula \( x = -\frac{b}{2a} \) in the standard form \( ax^2 + bx + c \).

Front

How can you determine if a parabola opens upward or downward?

Back

A parabola opens upward if the coefficient of \( a \) in the standard form \( ax^2 + bx + c \) is positive, and downward if \( a \) is negative.

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is a vertical line that divides the parabola into two mirror images. It can be found using the formula \( x = -\frac{b}{2a} \).

Front

What does it mean if a quadratic equation has no real solutions?

Back

It means that the graph of the quadratic does not intersect the x-axis, indicating that the discriminant \( b^2 - 4ac < 0 \).

Front

What is the quadratic parent function?

Back

The quadratic parent function is \( y = x^2 \). It is the simplest form of a quadratic function.

Front

What is the standard form of a quadratic function?

Back

The standard form of a quadratic function is \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

Front

What is the significance of the 'b' term in a quadratic function?

Back

The 'b' term affects the position of the vertex and the axis of symmetry of the parabola.

Front

How do you find the y-intercept of a quadratic function?

Back

The y-intercept can be found by evaluating the function at \( x = 0 \), which gives \( y = c \) in the standard form \( y = ax^2 + bx + c \).

Front

What is the effect of changing the 'a' value in a quadratic function?

Back

Changing the 'a' value affects the width and direction of the parabola. A larger absolute value of 'a' makes the parabola narrower, while a smaller absolute value makes it wider.

Front

What is the discriminant of a quadratic equation?

Back

The discriminant is given by \( b^2 - 4ac \) and determines the nature of the roots: if positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.