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The vertex of a parabola is like the peak or valley of a roller coaster. It's that special point where the parabola changes direction. For any parabola given by the quadratic equation \( y = ax^2 + bx + c \), the vertex can be pinpointed through a specific formula. We use \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex. In our case, substituting 1 for \( a \) and 3 for \( b \), we calculate \( x = -\frac{3}{2} \).Now, to find the corresponding y-value, plug \( x = -\frac{3}{2} \) back into the equation \( y = x^2 + 3x \). Doing so gives us the y-coordinate of \( -\frac{9}{4} \). Therefore, our vertex sits at \( \left(-\frac{3}{2}, -\frac{9}{4}\right) \). This is the lowest point on our parabola because the "a" value in our equation is positive, indicating it opens upwards.

Intercepts are the points where the graph of the parabola meets the axes. They serve as helpful landmarks for graphing functions.

• Y-intercept: To find the y-intercept, where the graph crosses the y-axis, simply set \( x = 0 \). For our equation, this gives \( y = 0 \), so the y-intercept is at \( (0,0) \).
• X-intercepts: When finding where the graph intersects the x-axis, set \( y = 0 \) and solve. For the quadratic equation \( x^2 + 3x = 0 \), factor it to get \( x(x+3) = 0 \). The solutions to this equation are \( x = 0 \) and \( x = -3 \), which means our x-intercepts are at \( (0, 0) \) and \( (-3, 0) \).

These intercepts give us key points through which we can draw our parabola, ensuring accuracy in our sketch.

When graphing a parabola, it helps to first gather all the necessary components: the vertex, and both x and y intercepts.
1. **Begin with the vertex**, which for our function is \( \left(-\frac{3}{2}, -\frac{9}{4}\right) \). Mark this point on your graph.2. **Add the intercepts.** Place the y-intercept at \( (0,0) \) and the x-intercepts at \( (0,0) \) and \( (-3,0) \).3. **Sketch the curve.** The parabola is symmetrical about the vertical line through the vertex, sometimes called the axis of symmetry, in this case at \( x = -\frac{3}{2} \). Our parabola opens upward since the "a" value in the equation is positive. Hence, it looks like a gentle "U" shape.Graphing the parabola gives a visual understanding of where it rises, drops, or flattens, aiding in a clearer comprehension of its properties.

The standard form of a quadratic equation is expressed as \( y = ax^2 + bx + c \). This structure provides essential clues about the parabola's shape and position.

• **"a" value:** This tells us the direction the parabola opens. If \( a > 0 \), the parabola opens upwards, resembling a smiling face. If \( a < 0 \), it opens downwards, like a frown. In our example, \( a = 1 \), so our parabola opens upwards.
• **"b" and "c" values:** These influence the position and shape of the parabola. The "b" factor, along with "a", helps locate the vertex's horizontal position, while "c" gives the y-intercept.

Understanding this form allows us to quickly determine critical attributes of the parabola, making it easier to graph and analyze. This knowledge is pivotal for solving real-world problems where quadratic functions appear.