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In a quadratic function, the x-intercepts are points where the graph crosses the x-axis. These occur when the value of the function, or "y," is zero. This means we set the equation of the quadratic to zero and solve for "x" to find these points.

In the original exercise, we have the quadratic equation in factored form as \( y = (x-2)(x+5) \). Here, the zero product property can be applied. According to this property, if the product of two expressions is zero, then at least one of the expressions must be zero as well.

So, setting each factor equal to zero gives:

• \(x-2 = 0\) which simplifies to \(x = 2\)
• \(x+5 = 0\) which simplifies to \(x = -5\)

This tells us that the x-intercepts of our parabola are \((2, 0)\) and \((-5, 0)\). These are the points where the curve touches or crosses the x-axis.

The vertex of a parabola is a key feature, marking the turning point of the graph. It's either the highest or lowest point, depending on whether the parabola opens upwards or downwards. For a quadratic equation in standard form \( ax^2 + bx + c \), the vertex can be calculated using the formula \( h = -\frac{b}{2a} \) for the x-coordinate.

Taking the quadratic from the original exercise which is \( y = x^2 + 3x - 10 \), we identify \(a = 1\) and \(b = 3\). Applying the formula:

• \( h = -\frac{3}{2*1} = -1.5 \)

We substitute \(x = -1.5\) back into the function to find the y-coordinate \(k\):

• \( y = (-1.5)^2 + 3(-1.5) - 10 = -6.25 \)

Thus, the vertex of the parabola is at \((-1.5, -6.25)\). This point is crucial as it helps in plotting and understanding the parabola's orientation and width on a graph.

Graphing quadratic equations involves sketching the curve known as a parabola on a coordinate plane. To graph any quadratic function, it is key to determine both the x-intercepts and the vertex first.

In our particular example, \( y = (x-2)(x+5) \), we have calculated the x-intercepts to be \((2, 0)\) and \((-5, 0)\). The vertex, which is also crucial, is found at \((-1.5, -6.25)\). Once these points are plotted on the graph, the shape of the parabola can be sketched by drawing a smooth, symmetrical curve through these points.

Additionally, since the quadratic coefficient (\(a\)) is positive \((a=1)\), it indicates that the parabola opens upwards. As you draw the curve, remember that parabolas are symmetrical about their vertex. Meaning, the left and right sides are mirror images.

• Start by marking the x-intercepts and vertex on the graph.
• Draw the axis of symmetry through the vertex, a vertical line for guidance.
• Connect the points with a smooth, U-shaped curve.

This process visually represents the solutions of the quadratic equation and helps understand the properties of quadratic functions.