[Resuelto]: h. y=4x^2+8x+2

Certainly! Let’s analyze the quadratic function

. We’ll explore its key features, including its vertex, intercepts, axis of symmetry, and graph characteristics.

1. Standard Form

The given quadratic function is in the standard form:

where:

2. Vertex

The vertex of a parabola described by a quadratic function can be found using the formulas:

Calculating the

-coordinate:

Substituting

into the original equation to find

Vertex:

3. Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex:

4. Y-Intercept

The y-intercept occurs when

Y-Intercept:

5. X-Intercepts (Roots)

To find the x-intercepts, set

and solve for

Using the quadratic formula:

Approximate X-Intercepts:

So, the x-intercepts are approximately

and

6. Direction of Opening

Since the coefficient of

(which is 4) is positive, the parabola opens upwards.

7. Graph Sketch

Here’s a summary of the key features to plot the graph:

Vertex:
Axis of Symmetry:
Y-Intercept:
X-Intercepts: Approximately and
Opening: Upwards

Steps to Sketch:

Plot the vertex at .
Draw the axis of symmetry as a dashed vertical line through .
Plot the y-intercept at .
Plot the x-intercepts at approximately and .
Since the parabola is symmetric, plot another point on the opposite side of the axis of symmetry for each x-intercept (e.g., reflect the y-intercept across the axis).
Draw a smooth curve through these points, opening upwards.

Graph Illustration:

While I can’t provide a visual graph here, you can use graphing tools like Desmos or GeoGebra to input the function

and visualize the parabola with the features described above.

8. Additional Features

Minimum Value: Since the parabola opens upwards, the vertex represents the minimum point of the function. Therefore, the minimum value of is -2, occurring at .
Range:
Domain: All real numbers ()

9. Summary

Vertex:
Axis of Symmetry:
Y-Intercept:
X-Intercepts: Approximately and
Direction: Opens upwards
Minimum Value: at
Range:
Domain: All real numbers

Feel free to ask if you need further clarification or additional information about this quadratic function!

Mind Expander

Did you know that this is a quadratic equation, which means it forms a parabola when graphed? In this case, the parabola opens upwards because the coefficient of

(which is 4) is positive. You can find the vertex of the parabola using the formula

, where

and

. This gives you the x-coordinate of the vertex, which is the minimum point on the graph.

If you’re looking to solve or graph this equation, keep an eye out for the common mistake of misidentifying the vertex! Remember that the vertex is not the same as the y-intercept. The y-intercept can be found simply by substituting

into the equation, leading you directly to the point (0, 2) on your graph!

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