 # Intersection of Quadratic Graphs

Consider the graphs of a quadratic function and a linear function on the same set of axes. cutting2 points of intersection $b^2-4ac \gt 0$ touching1 point of intersection $b^2-4ac = 0$ missingno points of intersection $b^2-4ac \lt 0$ In the graphs meet, the coordinates of the points of intersection of the graphs can be found [...] Example 1 Find the equation of the quadratic graph. Show Solution \begin{align} \displaystyle y &= a(x+2)(x+1) \\ 2 &= a(0+2)(0+1) &\text{substitute }(0,2)\\ 2 &= 2a \\ a &= 1 \\ y &= 1(x+2)(x+1) \\ \therefore y &= x^2+3x+2 \\ \end{align} Example 2 Find the equation of the quadratic graph. Show Solution \begin{align} [...] # Quadratic Graphs by Completing the Square y=(x-a)^2+b y=(x-a)^2+b Example 1 Draw the graph of y=(x-1)^2+2. Show Solution The vertex is (1,2) and the graph is concave up. Example 2 Draw the graph of y=(x-1)^2-2. Show Solution The vertex is (1,-2) and the graph is concave up. Example 3 Draw the graph of y=(x+1)^2+2. Show Solution The vertex is (-1,2) and the graph [...] # Vertex of a Parabola  y=(x+2)^2+1   y=-(x-2)^2-1  y=-(x+2)^2-1   y=-(x-2)^2+1   y=-(x+2)^2+1   y=(x-2)^2-1   y=(x+2)^2-1   y=(x-2)^2+1  Example 1 Find the vertex of y=x^2 + 4x +2. Show Solution \( \begin{align} \displaystyle y &= x^2 + 4x +2 \\ &= x^2 + 4x + 4 - 2\\ &= (x+2)^2-2 [...] # Axis of Symmetry x=-\dfrac{b}{2a} The equation of the axis of symmetry of y=ax^2+bx+c is x=-\dfrac{b}{2a}. Example 1 Find the equation of the axis of symmetry of y=x^2+4x-2. Show Solution \( \begin{align} \displaystyle x &= -\dfrac{b}{2a} \\ &= -\dfrac{4}{2 \times 1} \\ \therefore x &= -2 \\ \end{align} Example 2 Find the equation of the axis of symmetry of [...] $x$-intercepts when $y=0$ $y$-intercepts when $x=0$ Example 1 Sketch the graphs of $y=x^2+x-2$ by stating $x$- and $y$-intercepts. Show Solution $x$-intercepts when $y=0$ \begin{align} \displaystyle x^2+x-2 &= 0 \\ (x+2)(x-1) &= 0 \\ x &= -2, 1 \\ \end{align} $y$-intercepts when $x=0$ \begin{align} \displaystyle y &= 0^2+0-2 \\ &= -2 \\ \end{align} [...] # Parabola Graph in Intercept Form y=a(x-b)(x-c) y=a(x-b)(x-c) Concave up for a \gt 0 Concave down for a \lt 0 Example 1 Draw the graph of y=(x-1)(x+2). Show Solution \( \begin{align} \displaystyle (x-1)(x+2) & = 0 \\ x-1 &= 0 \text{ or } x+2 = 0 \\ \therefore x &= 1 \text{ or } x = -2 \\ \end{align} Example 2 [...] # Quadratic Graph of $y=(x-a)^2$ Completed Square Form

$$y=(x-a)^2+b$$ Example 1 Draw the graph of $y=(x-1)^2+2$. Show Solution The vertex is $(1,2)$ and the graph is concave up. Example 2 Draw the graph of $y=(x-1)^2-2$. Show Solution The vertex is $(1,-2)$ and the graph is concave up. Example 3 Draw the graph of $y=(x+1)^2+2$. Show Solution The vertex is $(-1,2)$ and the graph [...]