In Exercises 17–38, 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+3x−10

Verified step by step guidance

1. Identify the quadratic function given: $f\left(x\right)=x^2+3x-10$.
2. Find the vertex of the parabola using the formula for the x-coordinate of the vertex: $x=-\frac{b}{\mathrm{2a}}$, where $a\; =\; 1$ and $b\; =\; 3$. Then substitute this x-value back into the function to find the y-coordinate of the vertex.
3. Determine the axis of symmetry, which is the vertical line that passes through the vertex. Its equation is $x\; =$ the x-coordinate of the vertex found in the previous step.
4. Find the y-intercept by evaluating $f(0)$. Find the x-intercepts by solving the quadratic equation $x^2\; +\; 3x\; -\; 10\; =\; 0$ using factoring, completing the square, or the quadratic formula.
5. Use the vertex and intercepts to sketch the graph of the parabola. From the graph, state the domain (all real numbers) and the range (based on the vertex's y-coordinate and the parabola opening direction).

Key Concepts

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. For a quadratic function in standard form f(x) = ax^2 + bx + c, the vertex's x-coordinate is found using -b/(2a). The vertex helps in sketching the graph and understanding the function's behavior.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is x = -b/(2a). This line is crucial for graphing because it shows the parabola's symmetry and helps locate points on the graph.

Domain and Range of a Quadratic Function

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex: if the parabola opens upward (a > 0), the range is all y-values greater than or equal to the vertex's y-coordinate; if it opens downward (a < 0), the range is all y-values less than or equal to the vertex's y-coordinate.