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The y-intercept of a quadratic function is the point where the graph crosses the y-axis. It occurs when the value of \(x\) is zero. For any quadratic equation in the form \(f(x) = ax^2 + bx + c\), the y-intercept is simply \(f(0) = c\). This is because when you substitute \(x = 0\) into the equation, the terms involving \(x\) disappear, leaving only the constant term.

In our example, the function is \(f(x) = 2x^2\). Thus, \(f(0) = 2(0)^2 = 0\), and the y-intercept is at the origin, \((0, 0)\). This means that the graph intersects the y-axis at this point. The y-intercept provides a starting point for graphing the parabola.

The axis of symmetry of a parabola is a vertical line that divides the graph into two mirror-image halves. It passes through the vertex of the parabola. For any quadratic function in standard form \(f(x) = ax^2 + bx + c\), the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\).

In the example \(f(x) = 2x^2\), the coefficients are \(a = 2\) and \(b = 0\). Plugging these into the formula gives \(x = -\frac{0}{2 \times 2} = 0\). Thus, the axis of symmetry is the line \(x = 0\), which is actually the y-axis in this case.

• Helps determine the location of the vertex.
• Ensures that the parabola is symmetric on either side of this line.

Understanding the axis of symmetry is key to graphing the parabola.

The vertex of a parabola is its highest or lowest point, depending on whether it opens upwards or downwards. It gives the graph its distinct U-shape. For a function in the form \(f(x) = ax^2 + bx + c\), the x-coordinate of the vertex is found using the axis of symmetry, as \(x = -\frac{b}{2a}\).

Once the x-coordinate is known, substitute it back into the function to find the y-coordinate. For \(f(x) = 2x^2\), we have already determined that \(x = 0\). Substituting \(x = 0\) back into the function, \(f(0) = 0\). Therefore, the vertex of this parabola is \((0, 0)\).

• Indicates the minimum or maximum point of the parabola.
• Essential for creating the table of values for graphing.

The vertex plays a pivotal role in understanding the shape and position of the parabola.

Graphing a parabola involves several steps that tie together the concepts of y-intercept, vertex, and axis of symmetry. Once these are found, you can plot the parabola on a graph. First, place the y-intercept and vertex on the graph. Then, choose a few points on either side of the vertex along the x-axis to help determine the shape.

For \(f(x) = 2x^2\), we can create a table by substituting different values of \(x\) and finding \(f(x)\). For example:

• \(x = -2\), \(f(x) = 8\)
• \(x = -1\), \(f(x) = 2\)
• \(x = 0\), \(f(0) = 0\)
• \(x = 1\), \(f(1) = 2\)
• \(x = 2\), \(f(2) = 8\)

Once these points are plotted, draw a smooth curve through them, ensuring it is symmetric about the axis of symmetry. Since \(a = 2\) is positive, the parabola opens upwards, forming a U-shape. This method helps visualize the quadratic function and understand its properties.