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The vertex of a quadratic function is a key point that defines its geometry. It is the highest or lowest point on the graph of a parabola, depending on whether it opens downward or upward. For the function given, \( f(x) = x^2 + 4x - 5 \), we identified the coefficients as \( a = 1 \), \( b = 4 \), and \( c = -5 \). To find the vertex, we use the vertex formula:

• \( h = -\frac{b}{2a} \)
• Substitute in \( b = 4 \) and \( a = 1 \) to get \( h = -2 \).

Once \( h \) is determined, calculate \( k \) by substituting back into the function:

• \( k = (-2)^2 + 4(-2) - 5 = -9 \)

Therefore, the vertex is located at \( (-2, -9) \). This point is crucial for graphing as it informs you of the parabola's lowest point when the graph opens upwards.

A parabola is the U-shaped curve that emerges in the graph of a quadratic function. Its shape is determined by the value of the coefficient \( a \) in the standard quadratic equation form \( ax^2 + bx + c \). Here, \( a = 1 \), which is positive, indicating the parabola opens upward. This orientation means that the ends of the parabola extend infinitely upwards, and the vertex denotes its lowest point. The axis of symmetry, which splits the parabola into mirror images, passes vertically through the vertex. In our example, it is the line \( x = -2 \), directly through \( (-2, -9) \). Understanding the parabola's opening direction is essential for sketching and interpreting problems related to quadratic functions.

X-intercepts are the points where the graph of a function crosses the x-axis. This occurs when \( f(x) = 0 \). For the quadratic function \( f(x) = x^2 + 4x - 5 \), we find the x-intercepts by solving the equation:

• \( x^2 + 4x - 5 = 0 \)

From factorization, we identify two numbers that multiply to \(-5\) and add up to \(4\):

• They are \(5\) and \(-1\).
• So, the factorization is \((x + 5)(x - 1) = 0\).

Solving these factors:

• \( x + 5 = 0 \) gives \( x = -5 \)
• \( x - 1 = 0 \) gives \( x = 1 \)

Thus, the x-intercepts are \((-5,0)\) and \((1,0)\). These points are vital for graphing as they indicate where the parabola crosses the x-axis.

The y-intercept is the point where a graph intersects the y-axis. For quadratic functions, it's found by evaluating the function at \( x=0 \). In our function, substitute \( x = 0 \) into \( f(x) = x^2 + 4x - 5 \), yielding:

• \( f(0) = 0^2 + 4\cdot0 - 5 = -5 \)

The y-intercept is hence \((0, -5)\). When sketching a parabola, this is a straightforward point to plot. It ensures that your graph aligns correctly with the y-axis and helps in building the rest of your graph by connecting it through this point to the vertex and x-intercepts.