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The y-intercept of a graph is a crucial concept in algebra, particularly when dealing with functions and graphs. It's the exact point where a graph crosses the y-axis. For any function described by the equation \(y = f(x)\), the y-intercept happens at the value of \(y\) when \(x = 0\). This means, wherever you want to find the y-intercept, you simply need to substitute \(x = 0\) into the equation.

Why does this work? Think about the y-axis itself. It's the vertical axis on a graph where every point has an x-coordinate of 0. So, solving for \(y\) when \(x = 0\) effectively tells you the height or depth of the curve at its point closest to the y-axis.

• For example, in the equation \(y = x^2 - 4x + 3\), substituting \(x = 0\) gives \(y = 0^2 - 4 \cdot 0 + 3 = 3\). This tells us the y-intercept is at (0, 3) on the graph.

In the context of the parabola in this exercise, the y-intercept can quickly show you where the curve crosses the y-axis, giving a starting point for sketching the graph formed by the quadratic equation.

Parabolas are unique and important shapes in the study of mathematics, especially when graphing quadratic equations. A parabola is the shape of the graph of a quadratic function, and it will always look like a U or an inverted U. This U shape is called concavity. Parabolas can open upwards (concave up) or downwards (concave down), depending on the sign of the squared term's coefficient.

For the equation \(y = x^2 - 4x + 3\), the parabola opens upwards because the coefficient of \(x^2\) is positive.

Parabolas are defined by key features, such as:

• Vertex: The highest or lowest point on the parabola, depending on its orientation.
• Axis of Symmetry: A vertical line through the vertex, showing that the parabola is mirrored on both sides.
• Y-intercept: Where the parabola crosses the y-axis, as explained earlier.

Understanding parabolas and their properties helps you sketch any quadratic graph and analyze it effectively. You can predict the overall shape and position on the graph just by knowing these characteristics.

Substitution is a fundamental method used in mathematics to simplify equations or solve them for specific variables. When we talk about finding the y-intercept of a quadratic equation, we apply the substitution method to find where the function crosses the y-axis.

Using substitution here refers to plugging in a known value for one variable, \(x\), to find another, \(y\). For the exercise, when determining the y-intercept, we substitute \(x = 0\) into the given quadratic equation \(y = x^2 - 4x + 3\).

• This results in \(y = 0^2 - 4 \cdot 0 + 3\), simplifying directly to \(y = 3\).

Substitution is not only used here but is also a powerful tool across numerous mathematical problems. It allows you to break down complex expressions into simpler parts by using known values or expressions to solve for unknowns. In the context of quadratic equations, substitution forms a foundational step in graphing the equation and determining its intercepts and key points.