Quadratic Functions

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What is a quadratic function?

A quadratic function is a type of polynomial function with the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

What is the graph of a quadratic function called?

The graph of a quadratic function is called a parabola.

What shape does the parabola take if a > 0?

If a > 0, the parabola opens upwards and looks like a 'U' shape.

What happens to the parabola if a < 0?

If a < 0, the parabola opens downwards and looks like an upside-down 'U' shape.

What is the vertex of a parabola?

The vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards.

How do you find the vertex of a quadratic function in standard form?

For the equation y = ax² + bx + c, the vertex can be found using the formula (-b/2a, f(-b/2a)).

What is the axis of symmetry in a parabola?

The axis of symmetry is a vertical line that passes through the vertex of the parabola, given by the equation x = -b/2a.

What are the roots of a quadratic equation?

The roots are the values of x that make the quadratic equation equal to zero, also known as the solutions or x-intercepts.

How can you find the roots of a quadratic function?

Roots can be found using the quadratic formula, factoring, or by completing the square.

What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the roots of ax² + bx + c = 0.

What does the discriminant tell us about the roots of a quadratic function?

The discriminant (b² - 4ac) indicates the nature of the roots: two real and distinct if positive, one real and repeated if zero, and two complex if negative.

How does changing the value of 'b' affect the parabola?

Changing 'b' affects the position of the vertex and the axis of symmetry but does not affect the direction the parabola opens.

What role does 'c' play in the quadratic equation y = ax² + bx + c?

The 'c' value represents the y-intercept of the parabola.

How can you express a quadratic function in vertex form?

A quadratic function in vertex form is expressed as y = a(x-h)² + k, where (h, k) is the vertex.

How do you convert a quadratic equation to vertex form?

To convert to vertex form, you complete the square to rewrite y = ax² + bx + c as y = a(x-h)² + k.




Test Your Knowledge

Select the correct option


1. What is the quadratic formula?

x = (-b ± 2a + c) / a

x = (-b ± √(b² - 4ac)) / (2a)

x = (b ± 2a) / (b² - 4ac)

x = (-b ± √(4ac - b²)) / a

2. What is the vertex of a parabola?

The intersection of the parabola with the axis of symmetry.

Where the parabola meets the x-axis.

The highest or lowest point on the parabola.

The leftmost or rightmost point on a parabola.

3. How can you find the roots of a quadratic function?

Using the cubic formula, completing the circle, or factoring.

Roots can be found using the quadratic formula, factoring, or by completing the square.

By deriving and solving for zero.

Applying the Pythagorean theorem.

4. What does the discriminant tell us about the roots of a quadratic function?

It tells us about the speed at which the parabola grows.

It determines the width of the parabola.

The discriminant (b² - 4ac) indicates the nature of the roots: two real and distinct if positive, one real and repeated if zero, and two complex if negative.

It helps find the slope of the tangent line to the parabola.

5. What is the graph of a quadratic function called?

The graph of a quadratic function is called a parabola.

A circular function graph.

A straight line graph.

A hyperbolic graph.