Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry.. $$ y=-10 x^{2}+5 x-3 $$

Short answer

The graph opens down, the vertex of the function is at (0.25, -3.375) and the equation of the axis of symmetry is \(x=0.25\).

Step-by-step solution

Determine the Opening Direction

The sign of the a parameter determines if the graph opens up or down. Here, the a parameter is -10, a negative value, which means the graph opens down.

Find the Vertex

The x-coordinate of the vertex (h) can be found using the formula \(-b/2a\), substituting a with -10 and b with 5. By calculating, the result will be \(h=(-5)/(-20)=0.25\). Then substitute this h value into the function \(y=-10x^{2}+5x-3\) to find the y-coordinate. The vertex coordinates are therefore (0.25, -3.375).

Find the Axis of Symmetry

The axis of symmetry for a quadratic function is always of the form \(x=h\), where h is the x coordinate of vertex. Here, \(h=0.25\), so the equation for the axis of symmetry is \(x=0.25\).

Key concepts

Graph of a Quadratic Function

The graph of a quadratic function, such as \(y = ax^2 + bx + c\), is a U-shaped curve called a parabola. Whether the parabola opens upward or downward depends on the coefficient \(a\). If \(a\) is positive, the graph opens up, and if \(a\) is negative, as in our function \(y = -10x^2 + 5x - 3\), the graph opens down.

The graph's width varies with the value of \(a\); a larger absolute value of \(a\) results in a narrower parabola, while a smaller absolute value results in a wider one. This width is important because it affects not only the visual appearance of the graph but also the function's rate of change. In our example, the value of \(a = -10\) suggests a steep and narrow parabola.

Vertex of a Quadratic Function

The vertex of a quadratic function is the highest or lowest point on the graph, known as the turning point. For the function \(y = -10x^2 + 5x - 3\), we determined the vertex using the formula \(h = -\frac{b}{2a}\). After substituting the values of \(a\) and \(b\) into the formula, we calculated the vertex's x-coordinate to be \(0.25\).

Once we found the x-coordinate, we substituted it back into the function to find the corresponding y-coordinate, calculated as \(-3.375\). Therefore, the vertex of our function is at the coordinates \((0.25, -3.375)\), which is the point around which the graph is symmetric and the direction of the graph changes.

Axis of Symmetry

The axis of symmetry for a quadratic function is a vertical line that divides the graph into two mirror-image halves. It passes through the vertex, thus having the equation \(x = h\), where \(h\) is the x-coordinate of the vertex.

In our specific function \(y = -10x^2 + 5x - 3\), the vertex was found to be at \(0.25\), so the equation for the axis of symmetry is \(x = 0.25\). The graph will reflect across this line, which means that points to the left of the axis of symmetry will have corresponding points to the right with the same y-values, establishing this line as a wall of mirroring for the parabolic shape of the quadratic function graph.