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To find where a graph crosses the x-axis, you need to understand the concept of the x-intercept. The x-intercept is the point where the graph of an equation meets the x-axis. At this point, the y-value is always zero because the point lies directly on the x-axis. So, our task is to find x-values where the equation is zero.For the equation given: \[ y = 4x - x^2 \] set the y to zero: \[ 0 = 4x - x^2 \]This transforms into a simple quadratic equation when rearranged:\[ x^2 - 4x = 0 \]Factor the equation:

• Take out the common factor \(x\): \(x(x - 4) = 0\)

Now, solve for x by setting each factor equal to zero:

• If \(x = 0\), we have one intercept at (0, 0).
• If \(x - 4 = 0\), solve to get \(x = 4\), another intercept at (4, 0).

These steps show that the graph crosses the x-axis at points (0, 0) and (4, 0). Each x-intercept represents a solution to the equation when y equals zero.

Next, let’s explore how to find the y-intercept of a graph. This is where the graph crosses the y-axis. At this point, the x-value will always be zero because the point lies directly on the y-axis.To find the y-intercept of the equation \( y = 4x - x^2 \), substitute \(x = 0\):\[ y = 4(0) - (0)^2 = 0 \]When x is zero, y is also zero. Thus, the point where the graph meets the y-axis is at (0, 0).In simpler terms:

• Set x to zero in the equation.
• Calculate y with this value.
• The y-intercept is the y-value when x is zero.

This indicates the graph intersects the y-axis at the point (0, 0). Anytime you're uncertain about a graph's starting point in terms of y, computing the y-intercept is a great place to start.

Quadratic equations are fundamental expressions in algebra that involve a squared term. They are usually in the form: \( ax^2 + bx + c = 0 \). The equation given here: \( y = 4x - x^2 \) can be rearranged into this standard form: \( -x^2 + 4x + 0 = 0 \).What makes quadratics interesting is the shape of their graphs, typically parabolas. These are upward or downward-curving shapes based on the sign of the \(x^2\) coefficient:

• If positive, the parabola opens upwards.
• If negative, like in our equation, it opens downwards.

When analyzing a quadratic equation graph:

• The solutions for \(x\) when \(y = 0\) give the x-intercepts.
• The point where the parabola starts to turn is called the vertex.
• The y-intercept is easily found by setting \(x = 0\).

Grasping these features gives insight into graph behaviors, making it easier to understand and predict outcomes based on diverse scenarios.