How to Graph a Quadratic Function? Quadratic Function, Vertex, Axis of Symmetry and Parabola

Summary

In this video, the speaker explains how to graph quadratic functions by walking through two examples. The first function is in vertex form, and the speaker identifies the vertex, axis of symmetry, and direction of the parabola's opening based on the value of 'a'. The second example involves using a formula to find the vertex and continues with graphing the function's key features, such as the y-intercept and axis of symmetry. The video is aimed at helping viewers understand the process of graphing quadratic functions step-by-step.

Takeaways

• 📝 The quadratic function discussed is in vertex form: y = a(x - h)² + k.
• 📍 The vertex can be identified as (h, k). In the first example, the vertex is (1, 3).
• 🔼 The value of 'a' determines the parabola's opening direction. If a > 0, the parabola opens upward; if a < 0, it opens downward.
• 📏 The axis of symmetry is x = h, and it divides the parabola into two equal parts.
• 📊 The y-intercept is found by setting x = 0 in the equation. In the first example, the y-intercept is (0, 5).
• 🔄 The parabola is symmetric around the axis of symmetry, meaning points on one side are mirrored on the other.
• 📉 In the second example, the equation is y = 2x² + 4x + 2. The vertex is found using the formula for h: -b/2a.
• 🧮 For the second equation, the vertex is (-1, 0), and the y-intercept is (0, 2).
• 🔼 The parabola opens upward in both examples because the value of 'a' is positive.
• 📐 The quadratic graph process includes identifying the vertex, axis of symmetry, y-intercept, and plotting points to sketch the parabola.

Q & A

What is the vertex form of a quadratic equation?

The vertex form of a quadratic equation is written as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

How do you identify the vertex from the equation y = 2(x - 1)^2 + 3?

In the equation y = 2(x - 1)^2 + 3, the vertex is (h, k) = (1, 3). The h-value is 1 (opposite of the sign inside the parentheses), and the k-value is 3.

What does the value of 'a' in the quadratic equation determine?

The value of 'a' determines the direction of the parabola's opening. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards.

What is the axis of symmetry in a parabola, and how is it calculated?

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It is calculated as x = h, where h is the x-coordinate of the vertex.

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

To find the y-intercept, set x = 0 in the equation and solve for y. For example, in the equation y = 2(x - 1)^2 + 3, plugging in x = 0 gives y = 5, so the y-intercept is (0, 5).

What is the significance of the axis of symmetry in graphing a parabola?

The axis of symmetry helps to create a mirror effect in the parabola, ensuring that points on either side of the axis are symmetrically placed.

How do you find the x-intercept(s) of a quadratic function?

To find the x-intercept(s), set y = 0 and solve for x. This may involve factoring or using the quadratic formula.

What is the role of the vertex in graphing a quadratic function?

The vertex is the highest or lowest point on the graph of the parabola and serves as a key reference point in plotting the curve.

What is the general effect of the 'a' value on the shape of the parabola?

The value of 'a' affects the width and direction of the parabola. If |a| is greater than 1, the parabola is narrower. If |a| is less than 1, the parabola is wider.

How is the formula for finding the vertex from the standard form y = ax^2 + bx + c derived?

The formula for finding the x-coordinate of the vertex from the standard form is derived as x = -b/2a. Once the x-coordinate is known, it can be substituted into the equation to find the y-coordinate.