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Understanding how to calculate the x-intercept of a quadratic function is crucial for analyzing its graph. The x-intercept(s), also known as the root(s) or zero(s) of the function, occur where the graph crosses the x-axis. This transpires where the quadratic function equals zero, meaning that for the function \( f(x) \), we solve the equation \( f(x) = 0 \).

For the given function \( f(x) = (x-7)^2 \), we set the function equal to zero and solve for \( x \). Since \( (x-7)^2 = 0 \), it indicates that \( x-7 \) must also be zero. Hence, the solution to this equation is \( x=7 \). This single solution means that the graph of this function touches and bounces off from the x-axis at the point (7,0), unlike typical parabolas that usually cross the x-axis at two points.

The y-intercept of a quadratic function is the point where the graph intersects the y-axis. To find the y-intercept, you can simply substitute \( x = 0 \) into the quadratic function and evaluate.

Using the function \( f(x) = (x-7)^2 \), we plug in \( x = 0 \) which gives us \( f(0) = (0-7)^2 = 49 \). Therefore, the y-intercept is at the point (0,49). It's a single point on the graph that represents the value of the function when \( x \) is zero. This information is particularly helpful when sketching the graph since it provides a starting anchor point on the y-axis.

At the heart of a quadratic function like \( f(x) = (x-7)^2 \) lies the quadratic equation. Quadratic equations are of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The function given is a specific form of a quadratic equation, called a perfect square because it can be written as \( (x - h)^2 = 0 \), where \( h \) is the x-coordinate of the vertex.

Quadratics can have either two, one, or no real-world solutions. Intuitively, this relates to the graph of the quadratic function intersecting the x-axis at two points, touching the x-axis at exactly one point (as in this specific case where only one intercept at \( x=7 \) is found) or not touching the x-axis at all.

To graph quadratic functions, it's essential to identify the function's shape, direction, and critical points, such as the vertex, x-intercepts, and y-intercept.

A quadratic function typically graphs into a parabola, which is symmetrical about a line known as the axis of symmetry. For \( f(x) = (x-7)^2 \), the vertex is at (7,0), which is also the x-intercept, and the axis of symmetry is the vertical line \( x = 7 \). The y-intercept at (0,49) gives another point through which the parabola passes.

Moreover, the \( a \) term in the general quadratic equation \( ax^2 + bx + c \) informs us about the direction of the parabola. If \( a > 0 \), like in this function where \( a = 1 \), the parabola opens upwards. Conversely, if \( a < 0 \), the parabola opens downwards. The combination of these elements gives us a full picture of the parabola, enabling us to sketch its graph accurately.