Question

Graph the function. Label the vertex, axis of symmetry, and \(x\)-intercepts. \(h(x)=2 x(x-3)\)

Short answer

The vertex of the function is (1.5, -2.25), the axis of symmetry is \(x=1.5\), and the \(x\)-intercepts are \(x = 0\) and \(x = 3\). See graph for more detail.

Step-by-step solution

Rewrite the function in vertex form

The given quadratic function \(h(x) = 2x(x - 3)\) is currently in factored form. To get it into vertex form, we need to distribute the 2x through the parentheses: \(h(x) = 2x^2 - 6x\).

Find the vertex

We can compute this by using the formula \(h = -\frac{b}{2a}\), where \(a = 2\) and \(b = -6\) from the standard form of the quadratic equation \(y = ax^2 + bx + c\). Plugging in these values gives the x-coordinate of the vertex as \(h = -(-6)/2(2) = 1.5\). Determine the y-coordinate of the vertex by substituting \(x = 1.5\) into the function \(h(x)\), giving: \(h(1.5) = 2*(1.5)^2 - 6*(1.5) = -2.25\). So the vertex of the parabola is (1.5, -2.25).

Find the axis of symmetry

The equation for the axis of symmetry is \(x = h\), where \(h\) is the x-coordinate of the vertex. So the equation for the axis of symmetry is \(x = 1.5\).

Find the x-intercepts

To find the x-intercepts, we set \(h(x) = 0\) and solve for \(x\). The original equation \(h(x) = 2x(x - 3)\) is already factored, which makes this step easier: \(0=2x(x - 3)\). Solving this gives the x-intercepts (roots) as \(x = 0\) and \(x = 3\). Brainstorming these results will confirm our previous steps.

Graph the function

Draw a Cartesian coordinate system. Mark the point representing the vertex (1.5, -2.25). Draw a dotted line to represent the axis of symmetry \(x = 1.5\). Mark the two x-intercepts (0,0) and (3,0). Draw a smooth curve to connect these points to form the graph of the function \(h(x) = 2x(x - 3)\). The curve should be a parabola opening upwards, as deduced by our analysis.

Key concepts

Vertex Form

Quadratic functions can be rewritten in what's known as the "vertex form" to make graphing easier. The vertex form of a quadratic function is given by:
\[h(x) = a(x-h)^2 + k\]where

• \(a\) represents the coefficient that dictates the direction and width of the parabola.
• \(h\) and \(k\) represent the coordinates of the vertex of the parabola. \((h, k)\) becomes an important point that helps in sketching the graph.

To convert the function \(h(x)=2x(x-3)\) into vertex form, we initially expand it to \(h(x)=2x^2-6x\). Using the formula for the x-coordinate of the vertex \(h=-\frac{b}{2a}\) gives us \(x=1.5\). Plugging this back into the equation provides the y-coordinate, resulting in the vertex \((1.5, -2.25)\). This vertex coordinates guide us in sketching the parabola.

Axis of Symmetry

The axis of symmetry in a quadratic graph is a vertical line that passes through the vertex, effectively dividing the parabola into two mirror-image halves. This concept greatly simplifies graph plotting as it provides clear guidance on how to draw the graph symmetrically.
For any parabola in the form of \(y=ax^2+bx+c\), the axis of symmetry can be found using the formula:

• \(x = -\frac{b}{2a}\)

In our example, using \(a=2\) and \(b=-6\) from \(h(x)=2x^2-6x\), we determine that the axis of symmetry is at \(x=1.5\). This vertical line is crucial to the graph, as it marks the point through which each parabola's curve reflects. The line \(x = 1.5\) assists in verifying symmetry once the entire function graph is drawn.

X-Intercepts

X-intercepts are key points where a graph intersects the x-axis, marking the values of \(x\) for which \(h(x)=0\). These points provide crucial insights into the roots of the function and help establish the basic shape of the graph.
For the function \(h(x)=2x(x-3)\), it's already in factored form, making it straightforward to find x-intercepts by setting each factor equal to zero:

• \(2x = 0\) gives \(x=0\)
• \((x-3)=0\) results in \(x=3\)

With x-intercepts at \(x=0\) and \(x=3\), these points \((0,0)\) and \((3,0)\) anchor the graph's curve. They outline where the function values become zero, giving the parabola its "roots." Knowing these intercepts helps check the accuracy of your sketch when creating a graph of the quadratic function.