Find the x-intercepts of the given function, f Then use the...

Find the x-intercepts of the given function, $f$ Then use the $x$ -intercepts to match each function with its graph. [The graphs are labeled $(a) \text { through }(f) .]$
$f(x)=2(x+2)^{2}+5(x+2)-3$

Key Concepts

X-intercepts

X-intercepts are the points where a function’s graph crosses the x-axis, which occur when the function's output is zero. Finding the x-intercepts typically involves setting the function equal to zero and solving the resulting equation. In the context of quadratic functions, this often leads to solving a quadratic equation.

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two and can be solved using several methods such as factoring, completing the square, or the quadratic formula. The choice of method may depend on the form of the quadratic expression, and sometimes substitutions (like transforming variables) are used to simplify the process.

Substitution Technique

This technique involves replacing a composite expression within an equation with a single variable to simplify the equation. For example, if a function has a form like f(x) = a(x+c)^2 + b(x+c) + d, letting u = x+c transforms the equation into a standard quadratic in u, which is easier to solve.

Graph Analysis of Quadratic Functions

Graph analysis of quadratic functions includes understanding elements such as x-intercepts, y-intercepts, vertex, and axis of symmetry. Matching the algebraic solutions, like the x-intercepts, to the correct graph involves identifying these key features on the graph.

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