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The vertex of a quadratic function is a key feature of its graph, which is a parabola. The vertex represents the highest or lowest point on the parabola, depending on the direction it opens.
To find the vertex of a quadratic function in the standard form \( f(x) = ax^2 + bx + c \), use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation.
In the function \( f(x) = x^2 + 2x - 3 \):

• \( a = 1 \)
• \( b = 2 \)
• \( c = -3 \)

Substitute \( b = 2 \) and \( a = 1 \) into the formula:

• \( x = -\frac{2}{2(1)} = -1 \)

Next, calculate the \( y \)-coordinate by substituting \( x = -1 \) back into the function:

• \( f(-1) = (-1)^2 + 2(-1) - 3 = -4 \)

Therefore, the vertex is \((-1, -4)\). This point shows us where the parabola changes direction, forming the "tip" or "bottom" of the graph depending on the parabola's orientation.

The direction of a parabola is determined by the sign of the coefficient \( a \) in the quadratic function \( f(x) = ax^2 + bx + c \). This sign affects whether the parabola opens upwards like a "U" or downwards like an upside-down "U".
If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward. This is important as it tells us where the vertex is relative to the rest of the graph:

• Upward opening parabola: vertex is the lowest point (minimum)
• Downward opening parabola: vertex is the highest point (maximum)

For \( f(x) = x^2 + 2x - 3 \), since \( a = 1 \) which is greater than zero, the parabola opens upward. This means that the vertex \((-1, -4)\) is the lowest point of the graph.

Intercepts are crucial points where the graph of the parabola crosses the axes. There are two types of intercepts:

• Y-intercept: The point where the graph intersects the y-axis (when \( x = 0 \)).
• X-intercepts: The points where the graph intersects the x-axis (when \( f(x) = 0 \)).

To find the y-intercept of \( f(x) = x^2 + 2x - 3 \), set \( x = 0 \):

• Calculate: \( f(0) = (0)^2 + 2(0) - 3 = -3 \)

So, the y-intercept is \((0, -3)\). This represents the point where the parabola crosses the y-axis.
For x-intercepts, set the quadratic equation equal to zero:

• Solve: \( x^2 + 2x - 3 = 0 \)

You can factor the expression:

• \( (x + 3)(x - 1) = 0 \)

Solving these factors gives the x-values:

• \( x = -3 \)
• \( x = 1 \)

Thus, the x-intercepts are \((-3, 0)\) and \((1, 0)\). These points indicate where the parabola crosses the x-axis, and they are key in sketching the graph accurately.