Problem 29 - College Algebra

Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-x^{2}+3 x-10$$

Step-by-Step Solution

Answer

The vertex of the quadratic function is at (1.5, -12.25). The x-intercepts are at \(x = 2\) and \(x = -5\) and the y-intercept is at \(y = -10\). The axis of symmetry is \(x = 1.5\). The graph of the function is a parabola that opens upward, intersecting the x-axis at \(x = 2\) and \(x = -5\), and intersecting the y-axis at \(y = -10\). The domain of the function is all real numbers, and the range is \([-12.25, +∞)\).

Step 1: Identify the vertex

The standard form of a quadratic function is \(f(x) = a(x-h)^2 + k\), where \((h,k)\) is the vertex of the parabola. This function can be rewritten as \(f(x) = (x-1.5)^2 -12.25\), which means the vertex is at \((1.5, -12.25)\).

Step 2: Find the intercepts

To find the x-intercepts, set \(f(x) = 0\) and solve for \(x\). Doing so gives: \(0 = x^2 + 3x - 10\), and by solving the equation we get two x-intercepts which are \(x = 2\) and \(x = -5\). The y-intercept of the function is obtained by setting \(x = 0\) into the equation, giving the point \(0, -10\).

Step 3: Determine the axis of symmetry

The axis of symmetry of a parabola represented by a quadratic function in standard form \(f(x) = a(x-h)^2 + k\) is \(x = h\). In this case, the axis of symmetry is \(x = 1.5\).

Step 4: Sketch the graph

Plot the vertex, intercepts, and axis of symmetry on a graph. Draw the parabola by having the function curve upward from the vertex \((1.5, -12.25)\), intersecting the x-axis at \(x = 2\) and \(x = -5\), and intersecting the y-axis at \(y = -10\).

Step 5: Determine the domain and range

For a quadratic function, the domain is all real numbers, and the range is determined by the vertex. If the parabola opens upward, the range is \([k, +∞)\), and if it opens downward, it is \((−∞, k]\). In this case the parabola opens upward, so the range is \([-12.25, +∞)\).

Key Concepts

Quadratic Function Vertex

The vertex of a quadratic function is a crucial aspect that defines the highest or lowest point on its graph, depending on whether the parabola opens upwards or downwards. In the standard form of a quadratic function, which is written as \( f(x) = a(x-h)^2 + k \), the vertex is given by the coordinate pair \( (h, k) \).

For example, let's consider the function \( f(x) = x^2 + 3x - 10 \). By rewriting it in vertex form as \( f(x) = (x-1.5)^2 -12.25 \), we can identify that the vertex is at \( (1.5, -12.25) \). Understanding the vertex's position is instrumental to graphing the parabola and determining its other properties such as symmetry and range.

X-Intercepts and Y-Intercepts

Intercepts are where the graph of a function crosses the axes. The x-intercepts (or zeros) can be found by setting \( f(x) = 0 \) and solving for \( x \). For the given function, solving \( 0 = x^2 + 3x - 10 \) yields the x-intercepts at \( x = 2 \) and \( x = -5 \).

The y-intercept occurs where the function crosses the y-axis, which happens when \( x = 0 \). Substituting that into our function, we find the y-intercept at \( y = -10 \). Graphing these intercepts provides anchor points helping students visualize the path of the parabola as it intercepts the Cartesian plane.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex of a parabola and divides it into two mirror images. For a quadratic function in the form \( f(x) = a(x-h)^2 + k \), the axis of symmetry is defined by the line \( x = h \).

In our given function \( f(x) = x^2 + 3x - 10 \), after converting it to vertex form, we identify the axis of symmetry as the line \( x = 1.5 \). This axis is an essential guide for graphing because it ensures that the parabola is symmetrical. When plotting the graph, students should use the axis of symmetry to check their work for symmetry.

Domain and Range of a Function

The domain of a function represents all the possible input values (x-values), while the range is the set of possible output values (y-values). For any quadratic function, the domain is always all real numbers, because you can select any value for \( x \) and find a corresponding \( y \).

However, the range is dependent on the direction in which the parabola opens and its vertex. For an upward-opening parabola like our function \( f(x) = x^2 + 3x - 10 \), the range starts at the vertex's \( y \)-coordinate and extends to positive infinity, which in this case is \( [-12.25, +\infty) \). Teaching students about domain and range helps them understand the limits of the function's values.