Problem 29 - Algebra and Trigonometry

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 is \((-1.5, -12.25)\), the intercepts are \((-5,0)\), \((2,0)\), and \((0,-10)\), the axis of symmetry is \(x=-1.5\), the domain is \(-\infty \leq x \leq \infty\), and the range is \(-12.25 \leq y < \infty\).

Step 1: Calculate the Vertex

The vertex of a parabola in the form \(f(x) = ax^{2} + bx + c\) can be found by using the formula \(-b/(2a)\) to get the x-coordinate. Here, \(a=1\), and \(b=3\) so the x-coordinate of the vertex is \(-3/(2*1)=-1.5\). Substitute \(-1.5\) into the given function to find the y-coordinate of the vertex: \(f(-1.5) = (-1.5)^{2} + 3*(-1.5) - 10 = -12.25\). So, the vertex is \((-1.5, -12.25)\).

Step 2: Calculate the Intercepts

To find the x-intercepts, set \(f(x)=0\) and solve for \(x\). Thus, \(0=x^{2}+3x-10\). This can be factored to \((x+5)(x-2)=0\). Setting each factor equal to zero gives the roots \(x=-5, 2\). Therefore, the x-intercepts are \((-5,0)\) and \((2,0)\). For the y-intercept, set \(x=0\) in the function, resulting in \(f(0) = -10\). So, the y-intercept is \((0,-10)\).

Step 3: Sketch the graph and identify the axis of symmetry

Start with the vertex \((-1.5, -12.25)\), mark it on the graph. Then plot the intercepts \((-5,0)\), \((2,0)\), and \((0,-10)\). Given that it's a quadratic equation with \(a > 0\), the graph opens upward. Make sure to join all points with a smooth curve which forms the parabola. The parabola is symmetrical with the axis of symmetry passing through the vertex, so the axis of symmetry is \(x=-1.5\).

Step 4: Determine the domain and range

The domain of a parabola is all real numbers because any x-value can be substituted into the function. Therefore, the domain is \(-\infty\leq x \leq \infty\). The range is all y-values above its vertex since the parabola opens upwards. Therefore, the range is \(-12.25 \leq y < \infty\).

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is critical when sketching quadratic functions. The vertex is the highest or lowest point on the graph of a parabola, depending on whether it opens upwards or downwards. In the context of the function f(x)=x^2+3x-10, we calculate the x-coordinate of the vertex by using the formula -b/(2a). For this function, our values are a=1 and b=3, resulting in an x-coordinate of -1.5.

We then substitute this x-coordinate back into the original function to calculate the corresponding y-coordinate, which in our example yields -12.25. Thus, the vertex of the parabola is (-1.5, -12.25). It's the pivotal point from which the parabola will either open upwards or downwards and serves as an anchor for sketching the entire graph.

Quadratic X-Intercepts

Quadratic x-intercepts are points where the parabola crosses the x-axis. These are also referred to as the roots or zeros of the function. To find them, we set the quadratic function to zero and solve for x. In our example, the equation f(x)=0 leads to x^2+3x-10=0.

Solving this, we factor to get (x+5)(x-2)=0. Each factor can then be set to zero: x+5=0 or x-2=0, resulting in roots x=-5 and x=2. These reveal the x-intercepts of the parabola to be (-5,0) and (2,0), respectively, which are essential in plotting the quadratic graph.

Axis of Symmetry

The axis of symmetry in a parabola is a vertical line that divides the graph into mirror images. For a function in standard form y=ax^2+bx+c, the axis of symmetry always passes through the vertex. To find this line, we use the x-coordinate of the vertex. For our function f(x)=x^2+3x-10, since the vertex's x-coordinate is -1.5, the equation of the axis of symmetry is x=-1.5.

When sketching a parabola, the axis of symmetry is useful as a guideline for ensuring that the curve is symmetrical. It also indicates that for every point on one side of the parabola, there is a corresponding point equidistant on the other side.

Domain and Range of a Function

The domain and range of a function are sets of possible input (x-values) and output (y-values), respectively. For quadratic functions, the domain is always all real numbers, as there is no restriction on x-values that can be plugged into the function. Thus, the function f(x)=x^2+3x-10 has a domain of (-fty, fty).

The range, on the other hand, depends on the direction in which the parabola opens. For a parabola that opens upwards, like our example, the range starts at the vertex and goes to infinity, which in this case is [-12.25, fty). Understanding the domain and range helps in grasping how the function behaves and what values we can expect as output for our inputs.