Problem 14 - Algebra 1

Question

Sketch the graph of the function. Label the vertex. $$ y=x^{2}+4 x+7 $$

Step-by-Step Solution

Answer

The vertex of the quadratic function \(y = x^{2} + 4x + 7\) is at \((-2, 3)\). The graph is a parabola opening upwards with the vertex at this point.

Step 1: Identify a, b, and c

In the given quadratic function \(y = x^{2} + 4x + 7\), compare it with the standard form \(y = ax^{2} + bx + c\) to identify values of a, b and c. Here, \(a = 1\), \(b = 4\), and \(c = 7\).

Step 2: Find the x-coordinate of the vertex

Use the formula \(-\frac{b}{2a}\) to find the x-coordinate of the vertex. Here, it's \(-\frac{4}{2*1}\) which simplifies to \(-2\).

Step 3: Find the y-coordinate of the vertex

Substitute \(-2\) into the original function \(y = x^{2} + 4x + 7\). This gives \(y = (-2)^2 + 4*(-2) + 7\) which simplifies to 3. So the vertex is \((-2, 3)\).

Step 4: Sketch the graph

Sketch a parabola with the vertex at \((-2, 3)\). Remember that because \(a > 0\), the parabola opens upwards.

Key Concepts

Vertex Form

The vertex form of a quadratic function provides a powerful way to understand and handle parabolas. It is expressed as \( y = a(x-h)^2 + k \). Here, \((h, k)\) is the vertex of the parabola. This version of the equation effectively tells us not only the position of the vertex but also speaks to the parabola’s orientation and width based on the value of \(a\).

Transitioning from standard form \(y = ax^2 + bx + c\) to vertex form can give you deeper insights into the graph's geometry.
To convert it, you may complete the square or use the vertex formula for finding the vertex, and substitute values back into the vertex form.

• The vertex formula, \(h=-\frac{b}{2a}\) and then substituting into the function to find \(k\).
• This information will grant you the exact vertex \((h, k)\).
• You can then graph the function more precisely starting from this point.

This makes it easier to determine reflections, shifts, and stretches based on \(a\) and \((h, k)\). Exploring these transformations helps in graphing and understanding the behavior of the parabola.

Graphing Parabolas

Graphing parabolas involves a clear understanding of their properties and how they are influenced by different parameters in the quadratic equation. For the equation \(y = x^2 + 4x + 7\), its graph is a parabola. Parabolas have several important features:

• Vertex: The turning point or "top" of the parabola. We've calculated it as \((-2, 3)\).
• Axis of Symmetry: A vertical line that passes through the vertex, which is \(x = -2\).
• Direction: Since the coefficient \(a\) is positive, the parabola opens upwards.
• Y-intercept: Where the parabola crosses the y-axis, here it's \(y = 7\) when \(x = 0\).

When graphing:

• Start by marking the vertex on the graph.
• Draw the axis of symmetry.
• Using the y-intercept and vertex, sketch the parabola's shape.
• Consider additional points by evaluating the function at different x-values if a more detailed curve is needed.

This process aids in understanding the impact of each term in the quadratic equation on the graph’s shape and position.

Quadratic Formula

The Quadratic Formula offers a direct method for solving quadratic equations, particularly when the equation does not factor easily. It is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula allows you to find the x-intercepts of a quadratic function, which are the values where the parabola crosses the x-axis. For the equation \(y = x^2 + 4x + 7\), the discriminant \(b^2 - 4ac\) becomes an essential piece of the puzzle. In our example:

• Calculate the discriminant: \(b^2 - 4ac = 16 - 28 = -12\).
• Since the discriminant is negative, no real x-intercepts exist, indicating the parabola does not cross the x-axis.

This quadratic formula tool is not only limited to finding intercepts. It also provides:

• Insights into the nature of roots: real or complex.
• The ways parabolas behave in terms of crossing or touching the x-axis.

Utilizing the Quadratic Formula is key in exploring deeper facets of quadratic functions beyond merely graphing.