Problem 27 For each quadratic function, (a)... [FREE SOLUTION] (2024)

Short Answer

vertex of a parabola

The vertex of a parabola represents the highest or lowest point on a quadratic graph, commonly known as the turning point. Understanding how to find the vertex is crucial as it offers insights into the function’s maximum or minimum value depending on the parabola’s direction. For a quadratic function in standard form, $$ax^2 + bx + c, $$ the vertex $$(h, k) $$ can be found using the formulas: $$h = -\frac{b}{2a} $$ and $$k = g(h). $$ By substituting $$h $$ back into the quadratic function, we can calculate $$k. $$ This will precisely locate the vertex on the graph.

axis of symmetry

The axis of symmetry denotes the vertical line that divides the parabola into two mirror images. This line runs through the vertex of the parabola. For a quadratic function $$g(x) = ax^2 + bx + c, $$ the axis of symmetry can be determined using the equation: $$x = -\frac{b}{2a}. $$ This value of $$x $$ is the same as the $$x $$-coordinate $$h $$ of the vertex. Therefore, once the vertex is identified, the same value can be used as the axis of symmetry, thereby simplifying the graphing of the parabola.

graphing quadratic functions

Graphing quadratic functions involves several steps to ensure accuracy. Start by finding the vertex and axis of symmetry.
Next, plot these on the graph. Choose additional points around the vertex, and substitute them into the function to find their coordinates. Plot these points as well. For the given function $$ g(x)=x^2+3x-10, $$ after determining the vertex $$(-1.5, -12.25) $$ and the axis of symmetry $$x = -1.5, $$ we plot those and choose other points like $$x = -3 $$ and $$x = 0. $$Calculate the corresponding $$ y $$-values, which are $$g(-3) = -10 $$ and $$g(0) = -10. $$ Connect these points with a smooth, U-shaped curve to complete the graph.

standard form of a quadratic equation

The standard form of a quadratic equation is written as $$ax^2 + bx + c, $$ where $$a, b, $$ and $$c $$ are constants. This form is essential for analyzing various properties of the quadratic function, such as the vertex and axis of symmetry. In this form, $$a $$ determines the direction and width of the parabola, $$b $$ affects the direction the parabola shifts horizontally, and $$c $$ represents the vertical shift of the parabola. Recognizing the coefficients enables us to apply formulas to find critical points and aids in graphing the quadratic function accurately.

plotting points

Plotting points is a method used to draw the graph of a quadratic function. Begin by defining the critical points, like the vertex and additional points around it. For $$g(x) = x^2 + 3x - 10, $$ identify points such as the vertex $$(-1.5, -12.25), $$ and other points such as $$(-3, -10) $$ and $$(0, -10). $$ Substitute these $$x $$-values into the function to find the corresponding $$y $$-values. Plot these points on the graph, then draw a smooth curve through them to complete the parabola. This approach will provide a clearer visual representation of the quadratic function.