Problem 92 Graph each function. $$ y=x^... [FREE SOLUTION] (2024)

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A quadratic equation is a second-degree polynomial equation that can be expressed in the standard form as \(y=ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is nonzero. When graphed, a quadratic equation creates a U-shaped curve called a parabola. This type of function is noteworthy because it models many real-world phenomena, like the path of projectiles.

The equation \(y=x^2-4x+8\) is a clear example of a quadratic equation. In this form, you can see that \(a=1\) (dictating that the parabola opens upwards), \(b=-4\), and \(c=8\). Recognizing these values is the first step in graphing, as they shape the parabola's width, direction, and position on the coordinate plane.

Plotting a quadratic function involves finding key points like the vertex, axis of symmetry, y-intercept, and sufficient points to ensure a smooth curve. It is these characteristics that make the study of quadratic equations fundamental for understanding more complex mathematical concepts.

The vertex of a parabola is the highest or lowest point of the graph, depending on whether it opens upwards or downwards. In the standard quadratic function \(y=ax^2+bx+c\), you can find the vertex by using the formula \(h=-\frac{b}{2a}\). Once you've calculated \(h\), you can find \(k\) which is the value of \(y\) when \(x=h\).

For instance, with the equation \(y=x^2-4x+8\), applying the formula gives us \(h=2\). Substituting this value into the original equation to find \(k\), we get \(k=4\), which means the vertex is located at \((2, 4)\). This point is crucial for graphing because it gives you a reference for the rest of the parabola. In curriculum and testing, understanding how to identify the vertex is essential because it can be used to solve for maximum or minimum values in real-world problems.

The axis of symmetry in a parabola is a vertical line that divides the graph into two mirror-images. For any quadratic equation of the form \(y=ax^2+bx+c\), the axis of symmetry always passes through the vertex, making the vertex's x-coordinate immensely important. The axis of symmetry's equation is \(x=h\) where \(h\) is the x-coordinate of the vertex.

In our case with \(y=x^2-4x+8\), we have already established the vertex at \((2, 4)\), so the axis of symmetry is the line \(x=2\). Graphically, this means any point on one side of \(x=2\) is reflected across to the other at an equal distance, ensuring the parabola's symmetry. This concept is beneficial for sketching parabolas and is frequently tested in algebra courses.

The y-intercept of a graph refers to the point where the equation crosses the y-axis. To find the y-intercept of a quadratic function, simply substitute \(x=0\) into the equation and solve for \(y\). This tells you the point where the parabola meets the y-axis, which is always at \((0, c)\) for any quadratic equation in standard form.

For the equation \(y=x^2-4x+8\), when we plug in \(x=0\), we end up with \(y=8\). Therefore, the y-intercept for this parabola is the point \((0, 8)\). This point is helpful when you begin to plot the parabola, as it establishes where the graph starts or ends on the y-axis. Understanding the y-intercept is also important for interpreting many mathematical models in both algebra and calculus.