vertex
The vertex is a crucial point on the graph of a quadratic function. It's either the highest or lowest point of the parabola. For the quadratic function given, \( f(x) = -x^2 - 6x \), we find the vertex using the formula \( x = -\frac{b}{2a} \). Plugging in the coefficients, we get \( x = -\frac{-6}{2(-1)} = -3 \). To find the y-coordinate of the vertex, we substitute \( x = -3 \) back into the function: \( f(-3) = -(-3)^2 - 6(-3) = -9 + 18 = 9 \). Therefore, the vertex is \( (-3, 9) \). This point helps determine the graph's shape and direction.
axis of symmetry
The axis of symmetry is an imaginary vertical line that passes through the vertex. It divides the parabola into two mirror images. The axis of symmetry for a quadratic function is given by \( x = -\frac{b}{2a} \). In this case, using the coefficients from \( f(x) = -x^2 - 6x \), we already established \( x = -3 \). Thus, the axis of symmetry is the line \( x = -3 \). This line helps in plotting and understanding the function's symmetry.
concavity
Concavity describes whether the parabola opens upwards or downwards. It is determined by the coefficient \( a \) in the quadratic function \( f(x) = ax^2 + bx + c \). If \( a \) is positive, the parabola opens upwards (concave up). If \( a \) is negative, the parabola opens downwards (concave down). For \( f(x) = -x^2 - 6x \), \( a = -1 \). Since \( a \) is negative, the graph of the function is concave down. This means the vertex is the highest point of the parabola.
y-intercept
The y-intercept is the point where the graph crosses the y-axis. To find it, set \( x = 0 \) and solve for \( f(x) \). For \( f(x) = -x^2 - 6x \): \
x-intercepts
X-intercepts are the points where the graph crosses the x-axis. These points are found by setting \( f(x) = 0 \). For \( f(x) = -x^2 - 6x \): \( 0 = -x^2 - 6x \). Factor out \( -x \): \( -x(x + 6) = 0 \). Thus, the solutions are \( x = 0 \) and \( x = -6 \), giving x-intercepts at \( (0, 0) \) and \( (-6, 0) \). These points are critical for plotting the parabola's direction and width.
domain
The domain of a quadratic function refers to all possible x-values. For any quadratic function, including \( f(x) = -x^2 - 6x \), the domain is all real numbers because a quadratic function continues indefinitely in both the positive and negative directions along the x-axis. Thus, the domain is \( (-\infty, \, \infty) \). This tells us that we can plug in any real number for x and get a corresponding y-value.
range
The range captures all possible y-values of a quadratic function. For the function \( f(x) = -x^2 - 6x \) which opens downwards, the maximum y-value is the y-coordinate of the vertex. The vertex \( (-3, 9) \) gives us \( y = 9 \) as the highest point. Thus, the range includes all y-values less than or equal to 9: \( (-\infty, \, 9] \). This indicates the parabola extends infinitely downwards from 9.
increasing and decreasing intervals
For quadratic functions, increasing and decreasing intervals are determined by the vertex and the direction of the parabola. Given our function \( f(x) = -x^2 - 6x \), which opens downwards, the function increases to the left of the vertex and decreases to the right of the vertex. Hence, it increases on \( (-\infty, \, -3) \) and decreases on \( (-3, \, \infty) \). This helps us understand where to expect the function's output to rise or fall as we move along the x-axis.
