Characteristics and Transformations of Quadratic Graphs

Front

What is the vertex of a quadratic graph?

Back

The vertex is the highest or lowest point of the graph, depending on its orientation. It is represented as a coordinate point (h, k) in the vertex form of a quadratic function, f(x) = a(x-h)² + k.

Front

What does the axis of symmetry represent in a quadratic graph?

Back

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and is given by the equation x = h, where (h, k) is the vertex.

Front

How does a vertical shift affect the graph of a quadratic function?

Back

A vertical shift moves the graph up or down. For example, f(x) = x² + k shifts the graph up by k units if k is positive and down by k units if k is negative.

Front

What is the effect of changing the coefficient 'a' in the quadratic function f(x) = ax²?

Back

Changing 'a' affects the width and direction of the parabola. If |a| > 1, the graph is narrower; if 0 < |a| < 1, the graph is wider. If 'a' is negative, the parabola opens downward.

Front

What is the parent function of a quadratic graph?

Back

The parent function of a quadratic graph is f(x) = x². It is the simplest form of a quadratic function and serves as the basis for transformations.

Front

Describe a horizontal shift in a quadratic function. For example, f(x) = (x - 3)².

Back

A horizontal shift moves the graph left or right. In f(x) = (x - 3)², the graph shifts 3 units to the right. If it were (x + 3)², it would shift 3 units to the left.

Front

What is the significance of the y-intercept in a quadratic graph?

Back

The y-intercept is the point where the graph crosses the y-axis. It can be found by evaluating f(0). For f(x) = ax² + bx + c, the y-intercept is (0, c).

Front

How do you find the x-intercepts of a quadratic function?

Back

The x-intercepts can be found by setting f(x) = 0 and solving for x. This can be done using factoring, completing the square, or the quadratic formula.

Front

What does it mean for a quadratic function to have two real roots?

Back

A quadratic function has two real roots if its graph intersects the x-axis at two points. This occurs when the discriminant (b² - 4ac) is positive.

Front

What is the effect of a negative 'b' value in the quadratic function f(x) = ax² + bx + c?

Back

A negative 'b' value can affect the position of the vertex and the direction of the parabola. It can also influence the symmetry of the graph.