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The vertex of a parabola is its peak or lowest point, depending on the orientation of the parabola. For the quadratic equation \( y = x^2 + 4x + 3 \), the parabola opens upwards because the coefficient \( a = 1 \) is positive. The formula to find the vertex \( (h, k) \) for any parabola given by \(y = ax^2 + bx + c\) is: \[ h = -\frac{b}{2a}, \, k = f(h) \]Let's calculate the vertex for this equation:

• \( h = -\frac{4}{2 \times 1} = -2 \)
• Substitute \( h = -2 \) into the equation to find \( k \)
• \( k = (-2)^2 + 4(-2) + 3 = -1 \)

Thus, the vertex of the parabola is at \((-2, -1)\). This point is crucial for graphing, as it provides a central pivot point for the whole graph. The vertex also helps determine the minimum value of the parabola in this case.

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. This line passes through the vertex. For our given function \( y = x^2 + 4x + 3 \), the axis of symmetry can be directly derived from the x-coordinate of the vertex.The equation for the axis of symmetry is given as: \[ x = h \]where \( h \) is the x-coordinate of the vertex. For our equation:

• From the vertex \((-2, -1)\), \( h = -2 \)
• Therefore, the axis of symmetry is: \( x = -2 \)

This axis helps to ensure that when graphing, you maintain an equal shape on either side of this line. It acts like a balance point, which is very helpful in sketching a symmetric and accurate parabola.

Intercepts are points where the parabola crosses the axes. They help in constructing a more accurate graph of the parabola because they provide reference points.**Y-Intercept:**- The y-intercept occurs where the parabola crosses the y-axis, which means \( x = 0 \).- For \( y = x^2 + 4x + 3 \): - Substitute \( x = 0 \) to find \( y \) value. - \( y = (0)^2 + 4(0) + 3 = 3 \) - So, the y-intercept is: \( (0, 3) \)**X-Intercepts:**- The x-intercepts are points where the graph touches or crosses the x-axis, hence \( y = 0 \).- Solve \( x^2 + 4x + 3 = 0 \) using factoring: - \( (x + 1)(x + 3) = 0 \) - Therefore, \( x = -1 \) and \( x = -3 \) - So, the x-intercepts are: \((-1, 0)\) and \((-3, 0)\)These intercepts offer precise starting and ending points when sketching your parabola.

Graphing a parabola can be straightforward once you understand its components, such as the vertex, axis of symmetry, and intercepts. These components guide where and how the parabola should curve on a coordinate plane.Steps for Graphing:

• **Start with the vertex:** Place this point, \((-2, -1)\), on your graph. This point will anchor the parabola.
• **Draw the axis of symmetry:** A vertical dashed line through \(x=-2\). This will help maintain the balance of the curve on two sides.
• **Plot the intercepts:** Mark the y-intercept \((0, 3)\) and the x-intercepts \((-1,0)\) and \((-3,0)\).

Once all these points are plotted, draw a smooth curve through these points, ensuring it is cyclical and edges upwards as you move away from the vertex on either side of the axis of symmetry. The intercepts and other points help to guide the opening structure of the parabola, ensuring accuracy in the graph's curvature.