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In a quadratic function, the vertex represents the highest or lowest point on a parabola. Finding the vertex of a parabola is crucial for understanding its shape and position. The quadratic function given is \( f(x) = x^2 - 2x - 15 \). The formula used to find the x-coordinate of the vertex is \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are coefficients from the quadratic expression \( ax^2 + bx + c \).
For this function:

• \( a = 1 \)
• \( b = -2 \)

Substituting these values gives \( x = -\frac{-2}{2 \times 1} = 1 \). To find the y-coordinate, plug \( x = 1 \) back into the equation: \[f(1) = 1^2 - 2(1) - 15 = -16\]Thus, the vertex of the parabola is \((1, -16)\). This tells us that the minimum point of the graph occurs at \( x = 1 \), and its minimum value is \(-16\).

Intercepts are points where the graph crosses the axes. These include the y-intercept and x-intercepts. Let's explore how to find them for our function \( f(x) = x^2 - 2x - 15 \).

Y-Intercept: This occurs when \( x = 0 \). Substitute \( x = 0 \) into the function:\[f(0) = 0^2 - 2(0) - 15 = -15\]Thus, the y-intercept is \((0, -15)\). This point is where the graph cuts the vertical axis.

X-Intercepts: These are found by setting \( f(x) = 0 \). The equation becomes:\[x^2 - 2x - 15 = 0\]Factoring gives:\[(x - 5)(x + 3) = 0\]So, \( x = 5 \) and \( x = -3 \). The graph intersects the x-axis at points \((5, 0)\) and \((-3, 0)\). Recognizing intercepts helps in sketching the overall path of the parabola.

The direction in which a parabola opens is determined by the sign of the coefficient \( a \) in the quadratic function's standard form \( ax^2 + bx + c \).

In the function \( f(x) = x^2 - 2x - 15 \):

• \( a = 1 \)

Since \( a \) is positive, the parabola opens upwards. This means the graph has a "U" shape, with the vertex being the lowest point on the graph. If \( a \) were negative, the graph would open downward, resembling an upside-down "U" shape. Understanding the direction of the parabola helps predict the general motion of the graph and where the values of the function increase or decrease.

Sketching quadratic functions involves plotting the vertex, intercepts, and considering the parabola's direction. Let's put this together for \( f(x) = x^2 - 2x - 15 \).

First, identify key points:

• Vertex at \((1, -16)\)
• Y-intercept at \((0, -15)\)
• X-intercepts at \((5, 0)\) and \((-3, 0)\)

Place these points on a coordinate plane. Begin by plotting the vertex, which is crucial as it provides a base for the parabola. Then mark the intercepts. Connect these dots ensuring symmetry around the vertex.
Remember, the graph opens upwards because \( a \) is positive. Use a gentle curve to represent the parabola ensuring the vertex is the lowest part, and both sides of the parabola rise indefinitely outwards. Sketching promotes understanding of how algebra translates into a visual representation, making patterns in functions more observable and understandable.