Quadratic Function Characteristics and Equation
The question in the image asks you to consider a quadratic function, \(f\), with the following characteristics:
1. Axis of symmetry is the \(y\)-axis,
2. The \(x\)-intercepts are \( (-4, 0) \) and \( (4, 0) \),
3. The graph cuts the \(y\)-axis at \(y = 16\).
Followed by two sub-questions:
1.1. It asks to sketch the graph of \(f\) on a system of axes. Clearly show ALL intersepts with the axes.
1.2. It asks to determine the equation of the graph in the form \(ax^2 + q = y\).
Let's address each part:
For 1.1, sketching the graph requires understanding those characteristics:
- The axis of symmetry being the \(y\)-axis implies that the parabola will be mirrored across the \(y\)-axis.
- The \(x\)-intercepts are the points where the graph crosses the \(x\)-axis, which are given as \( (-4, 0) \) and \( (4, 0) \). These are the roots of the quadratic equation.
- The graph cuts the \(y\)-axis at \(y = 16\), which means the \(y\)-intercept is at the point \( (0, 16) \).
Plotting these points, starting with the \(x\)-intercepts at \(-4\) and \(4\), and the \(y\)-intercept at \(16\), you'll draw a "U"-shaped parabola that opens upwards.
For 1.2, to find the equation of the parabola, you can use the vertex form of a quadratic equation, \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. For this parabola, the vertex is on the \(y\)-axis, so \(h = 0\). Since the axis of symmetry is the \(y\)-axis, the vertex \(h\) is 0. Thus, the equation simplifies to \(y = ax^2 + k\).
The \(x\)-intercepts tell us the roots of the quadratic equation are \(x = -4\) and \(x = 4\). Therefore, the factored form of the quadratic is
\[ y = a(x + 4)(x - 4). \]
Expanding this gives us
\[ y = a(x^2 - 16). \]
We are also told the \(y\)-intercept is \(y = 16\), so when \(x = 0\):
\[ 16 = a(0 - 16), \]
which gives
\[ 16 = -16a, \]
Hence,
\[ a = -1. \]
So the equation of the parabola is:
\[ y = -1(x^2 - 16), \]
which simplifies to
\[ y = -x^2 + 16. \]
The question asks for a form where \(ax^2 + q = y\), so this is
\[ -x^2 + 16 = y, \]
which, when written in the requested form, is
\[ y = -x^2 + 16. \]
