https://file.aiquickdraw.com/mathbot/file/17272265875233h06cgcb.jpg

Determine the following: 1) Extrema 2) line of symmetry 3) x- intercepts 4) domain 5) tange show your work

Let's analyze the given function from the image and determine the required properties.

The equation from the graph is:

\[
y = (2x)^2 - 6x + 56
\]

which simplifies to:

\[
y = 4x^2 - 6x + 56
\]

Now let's solve each part one by one:

### 1) **Extrema (Vertex/Minimum Point)**

To find the extremum (the vertex), we can use the vertex formula for a quadratic equation \( y = ax^2 + bx + c \). The x-coordinate of the vertex is:

\[
x_v = -\frac{b}{2a}
\]

Here, \(a = 4\), \(b = -6\), and \(c = 56\).

\[
x_v = -\frac{-6}{2 \times 4} = \frac{6}{8} = \frac{3}{4}
\]

Now, to find the y-coordinate, substitute \(x_v = \frac{3}{4}\) into the equation:

\[
y_v = 4\left(\frac{3}{4}\right)^2 - 6\left(\frac{3}{4}\right) + 56
\]

\[
y_v = 4 \times \frac{9}{16} - 6 \times \frac{3}{4} + 56
\]

\[
y_v = \frac{36}{16} - \frac{18}{4} + 56 = 2.25 - 4.5 + 56 = 53.75
\]

Thus, the vertex is:

\[
\left(\frac{3}{4}, 53.75\right)
\]

This is the **minimum** point (since the parabola opens upwards).

### 2) **Line of Symmetry**

The line of symmetry passes through the x-coordinate of the vertex. Therefore, the equation of the line of symmetry is:

\[
x = \frac{3}{4}
\]

### 3) **X-intercepts**

To find the x-intercepts, we set \( y = 0 \) and solve for \( x \):

\[
4x^2 - 6x + 56 = 0
\]

This quadratic equation does not have real solutions because the discriminant is negative:

\[
\Delta = b^2 - 4ac = (-6)^2 - 4(4)(56) = 36 - 896 = -860
\]

Since \(\Delta < 0\), there are **no real x-intercepts**.

### 4) **Domain**

The domain of a quadratic function is always all real numbers:

\[
\text{Domain: } (-\infty, \infty)
\]

### 5) **Tangent Line at \(x = 1\)**

To find the tangent line at \(x = 1\), we first need to find the derivative of the function \(y = 4x^2 - 6x + 56\). The derivative is:

\[
\frac{dy}{dx} = 8x - 6
\]

Now, we evaluate the derivative at \(x = 1\) to find the slope of the tangent line:

\[
\frac{dy}{dx} \bigg|_{x=1} = 8(1) - 6 = 2
\]

The slope of the tangent line at \(x = 1\) is 2. Now, we find the y-coordinate at \(x = 1\):

\[
y(1) = 4(1)^2 - 6(1) + 56 = 4 - 6 + 56 = 54
\]

The equation of the tangent line is given by:

\[
y - y_1 = m(x - x_1)
\]

where \(m = 2\), \(x_1 = 1\), and \(y_1 = 54\). Thus, the equation of the tangent line is:

\[
y - 54 = 2(x - 1)
\]

\[
y = 2x + 52
\]

### Summary of Results:
1. **Extremum (minimum)**: \(\left(\frac{3}{4}, 53.75\right)\)
2. **Line of symmetry**: \(x = \frac{3}{4}\)
3. **X-intercepts**: None (no real x-intercepts)
4. **Domain**: \( (-\infty, \infty) \)
5. **Tangent at \(x = 1\)**: \( y = 2x + 52 \)

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How can we determine the concavity of a quadratic function?
2. What does the discriminant tell us about the roots of a quadratic equation?
3. How can we find the y-intercept of this function?
4. What role does the coefficient of \(x^2\) play in the shape of a parabola?
5. How do you find the points of intersection between two quadratic functions?

**Tip:** For quadratic functions, the axis of symmetry always passes through the vertex of the parabola.

Determine the following: 1) Extrema 2) Line of symmetry 3) x-intercepts 4) Domain 5) Tangent at x=1. Show your work.

Explore a step-by-step solution to a quadratic function, y = 4x^2 - 6x + 56, covering the vertex, symmetry, x-intercepts, domain, and tangent line calculation at x = 1. Learn key formulas like the quadratic equation, vertex formula, and the process for calculating tangents and discriminants.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation