https://file.aiquickdraw.com/mathbot/file/1730064508793wy63bmce.jpg

The questions in the image ask about quadratic functions and consist of the following tasks:

1. For the quadratic function \( y = 2x^2 - 8x + 4 \):
   - a) Complete the square on this function.
   - b) Sketch the function.
   - c) Find the vertex.

2. Sketch each quadratic function in the fourth quadrant:
   - i) \( y = 2x^2 + 12x + 1 \)
   - ii) \( y = 2x^2 - 6x + 15 \)

### Let's solve each part.

#### 1. For \( y = 2x^2 - 8x + 4 \):

##### Part a) Completing the Square
Rewrite the function in the form \( y = a(x - h)^2 + k \), where \((h, k)\) represents the vertex of the parabola.
1. Factor out the coefficient of \(x^2\) (which is 2) from the \(x\)-terms:
   \[
   y = 2(x^2 - 4x) + 4
   \]
2. Complete the square inside the parentheses:
   - Take half of \(-4\), square it to get \(4\), and add and subtract this value inside the parentheses.
   \[
   y = 2(x^2 - 4x + 4 - 4) + 4
   \]
   \[
   = 2((x - 2)^2 - 4) + 4
   \]
3. Expand and simplify:
   \[
   y = 2(x - 2)^2 - 8 + 4
   \]
   \[
   = 2(x - 2)^2 - 4
   \]
So, the function in vertex form is \( y = 2(x - 2)^2 - 4 \).

##### Part b) Sketch the Function
The parabola opens upwards (since the coefficient of \((x - 2)^2\) is positive) with a vertex at \((2, -4)\). You can sketch it by marking the vertex and plotting a few additional points on either side of \(x = 2\).

##### Part c) Find the Vertex
From the vertex form \( y = 2(x - 2)^2 - 4 \), the vertex is at \((2, -4)\).

#### 2. Sketch each quadratic function in the fourth quadrant:

##### i) For \( y = 2x^2 + 12x + 1 \):
   - Complete the square or use other methods to find the vertex, then sketch the function in the fourth quadrant if it intersects that region.

##### ii) For \( y = 2x^2 - 6x + 15 \):
   - Similarly, complete the square or use other methods to determine the vertex and the orientation of the parabola, then sketch it in the fourth quadrant if applicable.

Would you like further details on any specific part or help with the sketching process?

---

Here are some related questions for additional practice:

1. How do you complete the square for a general quadratic function?
2. How can you determine if a parabola opens upwards or downwards?
3. What role does the vertex form of a quadratic function play in graphing?
4. How do you find the axis of symmetry of a quadratic function?
5. How does the discriminant affect the nature of the roots of a quadratic equation?

**Tip:** When graphing a quadratic function, always identify the vertex and direction (up or down) first to understand the parabola’s shape.

1. A quadratic function is given as y = 2x^2 - 8x + 4
   a) Complete the square on this function
   b) Sketch the function
   c) Find the vertex

2. Sketch each quadratic function in the fourth quadrant:
   i) y = 2x^2 + 12x + 1
   ii) y = 2x^2 - 6x + 15

Learn how to complete the square, find the vertex, and sketch quadratic functions, with examples like y = 2x^2 - 8x + 4. This problem covers algebraic methods and graphing skills for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation